Hypoid gear design method and hypoid gear

ABSTRACT

A degree of freedom of a hypoid gear is improved. An instantaneous axis in a relative rotation of a gear axis and a pinion axis, a line of centers, an intersection between the instantaneous axis and the line of centers, and an inclination angle of the instantaneous axis with respect to the rotation axis of the gear are calculated based on a shaft angle, an offset, and a gear ratio of a hypoid gear. Based on these variables, base coordinate systems are determined, and the specifications are calculated using these coordinate systems. For the spiral angles, pitch cone angles, and reference circle radii of the gear and pinion, one of the values for the gear and the pinion is set and a design reference point is calculated. Based on the design reference point and a contact normal of the gear, specifications are calculated. The pitch cone angle of the gear or the pinion can be freely selected.

The present application is a divisional application of U.S. application Ser. No. 14/508,422, filed on Oct. 7, 2014, which is a divisional application of U.S. application Ser. No. 13/054,323, filed Feb. 3, 2011, which is a National Stage Entry of PCT/JP2009/063234, filed Jul. 16, 2009, which claims priority to each of JP 2009-111881, filed May 1, 2009, JP 2008-280558, filed Oct. 30, 2008, and JP 2008-187965, filed Jul. 18, 2008. The disclosures of each of the above applications are hereby incorporated by reference in their entireties.

TECHNICAL FIELD

The present invention relates to a method of designing a hypoid gear.

BACKGROUND ART

A design method of a hypoid gear is described in Ernest Wildhaber, Basic Relationship of Hypoid Gears, American Machinist, USA, Feb. 14, 1946, p. 108-111 and in Ernest Wildhaber, Basic Relationship of Hypoid Gears II, American Machinist, USA, Feb. 28, 1946, p. 131-134. In these references, a system of eight equations is set and solved (for cone specifications that contact each other) by setting a spiral angle of a pinion and an equation of a radius of curvature of a tooth trace, in order to solve seven equations with nine variables which are obtained by setting, as design conditions, a shaft angle, an offset, a number of teeth, and a ring gear radius. Because of this, the cone specifications such as the pitch cone angle Γ_(gw) depend on the radius of curvature of the tooth trace, and cannot be arbitrarily determined.

In addition, in the theory of gears in the related art, a tooth trace is defined as “an intersection between a tooth surface and a pitch surface”. However, in the theory of the related art, there is no common geometric definition of a pitch surface for all kinds of gears. Therefore, there is no common definition of the tooth trace and of contact ratio of the tooth trace for various gears from cylindrical gears to hypoid gears. In particular, in gears other than the cylindrical gear and a bevel gear, the tooth trace is not clear.

In the related art, the contact ratio m_(f) of tooth trace is defined by the following equation for all gears. m _(f) =F tan ψ₀ /p where, p represents the circular pitch, F represents an effective face width, and ψ₀ represents a spiral angle.

Table 1 shows an example calculation of a hypoid gear according to the Gleason method. As shown in this example, in the Gleason design method, the tooth trace contact ratios are equal for a drive-side tooth surface and for a coast-side tooth surface. This can be expected because of the calculation of the spiral angle ψ₀ as a virtual spiral bevel gear with ψ₀=(ψ_(pw)+ψ_(gw))/2 (refer to FIG. 9).

The present inventors, on the other hand, proposed in Japanese Patent No. 3484879 a method for uniformly describing the tooth surface of a pair of gears. In other word, a method for describing a tooth surface has been shown which can uniformly be used in various situations from a pair of gears having parallel axes, which is the most widely used configuration, to a pair of gears whose axes do not intersect and are not parallel with each other (skew position).

There is a desire to determine the cone specifications independent from the radius of curvature of the tooth trace, and to increase the degree of freedom of the design.

In addition, in a hypoid gear, the contact ratio and the transmission error based on the calculation method of the related art are not necessarily correlated to each other. Of the contact ratios of the related art, the tooth trace contact ratio has the same value between the drive-side and the coast-side, and thus the theoretical basis is brought into question.

An advantage of the present invention is that a hypoid gear design method is provided which uses the uniform describing method of the tooth surface described in JP 3484879, and which has a high degree of freedom of design.

Another advantage of the present invention is that a hypoid gear design method is provided in which a design reference body of revolution (pitch surface) which can be applied to the hypoid gear, the tooth trace, and the tooth trace contact ratio are newly defined using the uniform describing method of the tooth surface described in JP 3484879, and the newly defined tooth trace contact ratio is set as a design index.

SUMMARY OF THE INVENTION

According to one aspect of the present invention, there is provided a design method of a hypoid gear wherein an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers v_(c) which is common to rotation axes of the first gear and the second gear, an intersection C_(s) between the instantaneous axis S and the line of centers v_(c), and an inclination angle Γ_(s) of the instantaneous axis S with respect to the rotation axis of the second gear are calculated based on a shaft angle Σ, an offset E, and a gear ratio i_(o) of a hypoid gear, basic coordinate systems C₁, C₂, and C_(s) are determined from these variables, and specifications are calculated based on the coordinate systems. In particular, the specifications are calculated by setting a common point of contact of pitch cones of the first gear and the second gear as a design reference point P_(w).

When an arbitrary point (design reference point) P_(w) is set in a static space, six cone specifications which are in contact at the point P_(w) are represented by coordinates (u_(cw), v_(cw), z_(cw)) of the point P_(w) based on a plane (pitch plane) S_(t) defined by a peripheral velocity V_(1w) and a peripheral velocity V_(2w) at the point P_(w) and a relative velocity V_(rsw). Here, the cone specifications refer to reference circle radii R_(1w) and R_(2w) of the first gear and the second gear, spiral angles ψ_(pw) and ψ_(gw) of the first gear and the second gear, and pitch cone angles γ_(pw) and Γ_(gw) of the first gear and the second gear. When three of these cone specifications are set, the point P_(w) is set, and thus the remaining three variables are also set. In other words, in various aspects of the present invention, the specifications of cones which contact each other are determined based merely on the position of the point P_(w) regardless of the radius of curvature of the tooth trace.

Therefore, it is possible to set a predetermined performance as a design target function, and select the cone specifications which satisfy the target function with a high degree of freedom. Examples of the design target function include, for example, a sliding speed of the tooth surface, strength of the tooth, and the contact ratio. The performance related to the design target function is calculated while the cone specification, for example the pitch cone angle Γ_(gw), is changed, and the cone specification is changed and a suitable value is selected which satisfies the design request.

According to one aspect of the present invention, an contact ratio is employed as the design target function, and there is provided a method of designing a hypoid gear wherein a pitch cone angle Γ_(gcone) of one gear is set, an contact ratio is calculated, the pitch cone angle Γ_(gcone) is changed so that the contact ratio becomes a predetermined value, a pitch cone angle Γ_(gw) is determined, and specifications are calculated based on the determined pitch cone angle Γ_(gw). As described above, the contact ratio calculated by the method of the related art does not have a theoretical basis. In this aspect of the present invention, a newly defined tooth trace and an contact ratio related to the tooth trace are calculated, to determine the pitch cone angle. The tooth surface around a point of contact is approximated by its tangential plane, and a path of contact is made coincident to an intersection of the surface of action (pitch generating line L_(pw)), and a tooth trace is defined as a curve on a pitch hyperboloid obtained by transforming the path of contact into a coordinate system which rotates with each gear. Based on the tooth trace of this new definition, the original contact ratio of the hypoid gear is calculated and the contact ratio can be used as an index for design. A characteristic of the present invention is in the definition of the pitch cone angle related to the newly defined tooth trace.

According to another aspect of the present invention, it is preferable that, in the hypoid gear design method, the tooth trace contact ratio is assumed to be 2.0 or more in order to achieve constant engagement of two gears with two or more teeth.

When the pitch cone angle Γ_(gw) is set to an inclination angle Γ_(s) of an instantaneous axis, the contact ratios of the drive-side and the coast-side can be set approximately equal to each other. Therefore, it is preferable for the pitch cone angle to be set near the inclination angle of the instantaneous axis. In addition, it is also preferable to increase one of the contact ratios of the drive-side or coast-side as required. In this process, first, the pitch cone angle is set at the inclination angle of the instantaneous axis and the contact ratio is calculated, and a suitable value is selected by changing the pitch cone angle while observing the contact ratio. It is preferable that a width of the change of the pitch cone angle be in a range of ±5° with respect to the inclination angle Γ_(s) of the instantaneous axis. This is because if the change is out of this range, the contact ratio of one of the drive-side and the coast-side will be significantly reduced.

More specifically, according to one aspect of the present invention, a hypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of a hypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i₀, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers v_(c) with respect to rotation axes of the first gear and the second gear, an intersection C_(s) between the instantaneous axis S and the line of centers v_(c), and an inclination angle Γ_(s) of the instantaneous axis S with respect to the rotation axis of the second gear, and determining coordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting three variables including one of a reference circle radius R_(1w) of the first gear and a reference circle radius R_(2w) of the second gear, one of a spiral angle ψ_(pw) of the first gear and a spiral angle ψ_(gw) of the second gear, and one of a pitch cone angle γ_(pw) of the first gear and a pitch cone angle Γ_(gw) of the second gear;

(d) calculating the design reference point P_(w), which is a common point of contact of pitch cones of the first gear and the second gear, and the three other variables which are not set in the step (c), based on the three variables which are set in the step (c);

(e) setting a contact normal g_(wD) of a drive-side tooth surface of the second gear;

(f) setting a contact normal g_(wC) of a coast-side tooth surface of the second gear; and

(g) calculating specifications of the hypoid gear based on the design reference point P_(w), the three variables which are set in the step (c), the contact normal g_(wD) of the drive-side tooth surface of the second gear, and the contact normal g_(wC) of the coast-side tooth surface of the second gear.

According to another aspect of the present invention, a hypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of a hypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i₀, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers v_(c) with respect to rotation axes of the first gear and the second gear, an intersection C_(s) between the instantaneous axis S and the line of centers v_(c), and an inclination angle Γ_(s) of the instantaneous axis S with respect to the rotation axis of the second gear, and determining coordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting three variables including one of a reference circle radius R_(1W) of the first gear and a reference circle radius R_(2w) of the second gear, one of a spiral angle ψ_(pw) of the first gear and a spiral angle ψ_(gw) of the second gear, and one of a pitch cone angle γ_(pw) of the first gear and a pitch cone angle Γ_(gw) of the second gear;

(d) calculating the design reference point P_(w), which is a common point of contact of pitch cones of the first gear and the second gear, and the three other variables which are not set in the step (c), based on the three variable which are set in the step (c);

(e) calculating a pitch generating line L_(pw) which passes through the design reference point P_(w) and which is parallel to the instantaneous axis S;

(f) setting an internal circle radius R_(2t) and an external circle radius R_(2h) of the second gear;

(g) setting a contact normal g_(wD) of a drive-side tooth surface of the second gear;

(h) calculating an intersection P_(0D) between a reference plane S_(H) which is a plane orthogonal to the line of centers v_(c) and passing through the intersection C_(s) and the contact normal g_(wD) and a radius R_(20D) of the intersection P_(OD) around a gear axis;

(i) calculating an inclination angle φ_(s0D) of a surface of action S_(wD) which is a plane defined by the pitch generating line L_(pw) and the contact normal g_(wD) with respect to the line of centers v_(c), an inclination angle ψ_(sw0D) of the contact normal g_(wD) on the surface of action S_(wD) with respect to the instantaneous axis S, and one pitch P_(gwD) on the contact normal g_(wD);

(j) setting a provisional second gear pitch cone angle Γ_(gcone), and calculating an contact ratio m_(fconeD) of the drive-side tooth surface based on the internal circle radius R_(2t) and the external circle radius R_(2h);

(k) setting a contact normal g_(wC) of a coast-side tooth surface of the second gear;

(l) calculating an intersection P_(0C) between the reference plane S_(H) which is a plane orthogonal to the line of centers v_(c) and passing through the intersection C_(s) and the contact normal g_(wC) and a radius R_(20c) of the intersection P_(0C) around the gear axis;

(m) calculating an inclination angle ϕ_(s0C) of a surface of action S_(wC) which is a plane defined by the pitch generating line L_(pw) and the contact normal g_(wC) with respect to the line of centers v_(c), an inclination angle ψ_(sw0C) of the contact normal g_(wC) on the surface of action S_(wC) with respect to the instantaneous axis S, and one pitch P_(gwC) on the contact normal g_(wC);

(n) setting a provisional second gear pitch cone angle Γ_(gcone), and calculating an contact ratio m_(fconeC) of the coast-side tooth surface based on the internal circle radius R_(2t) and the external circle radius R_(2h);

(o) comparing the contact ratio m_(fconeD) of the drive-side tooth surface and the contact ratio m_(fconeC) of the coast-side tooth surface, and determining whether or not these contact ratios are predetermined values;

(p) when the contact ratios of the drive-side and the coast-side are the predetermined values, replacing the provisional second gear pitch cone angle Γ_(gcone) with the second gear pitch cone angle Γ_(gw) obtained in the step (c) or in the step (d);

(q) when the contact ratios of the drive-side and the coast-side are not the predetermined values, changing the provisional second gear pitch cone angle Γ_(gcone) and re-executing from step (g);

(r) re-determining the design reference point P_(w), the other one of the reference circle radius R_(1w) of the first gear and the reference circle radius R_(2w) of the second gear which is not set in the step (c), the other one of the spiral angle ψ_(pw) of the first gear and the spiral angle ψ_(gw) of the second gear which is not set in the step (c), and the first gear pitch cone angle γ_(pw) based on the one of the reference circle radius R_(1w) of the first gear and the reference circle radius R_(2w) of the second gear which is set in the step (c), the one of the spiral angle ψ_(pw) of the first gear and the spiral angle ψ_(gw) of the second gear which is set in the step (c), and the second gear pitch cone angle Γ_(gw) which is replaced in the step (p), and

(s) calculating specifications of the hypoid gear based on the specifications which are set in the step (c), the specifications which are re-determined in the step (r), the contact normal g_(wD) of the drive-side tooth surface of the second gear, and the contact normal g_(wC) of the coast-side tooth surface of the second gear.

According to another aspect of the present invention, a hypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of a hypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i₀, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers v_(c) with respect to rotation axes of the first gear and the second gear, an intersection C_(s) between the instantaneous axis S and the line of centers v_(c), and an inclination angle Γ_(s) of the instantaneous axis S with respect to the rotation axis of the second gear, to determine coordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting three variables including one of a reference circle radius R_(1w) of the first gear and a reference circle radius R_(2w) of the second gear, one of a spiral angle ψ_(pw) of the first gear and a spiral angle ψ_(gw) of the second gear, and one of a pitch cone angle γ_(pw) of the first gear and a pitch cone angle Γ_(gw) of the second gear;

(d) calculating the design reference point P_(w), which is a common point of contact of pitch cones of the first gear and the second gear, and the three other variables which are not set in the step (c), based on the three variables which are set in the step (c);

(e) calculating a pitch generating line L_(pw) which passes through the design reference point P_(w) and which is parallel to the instantaneous axis S;

(f) setting an internal circle radius R_(2t) and an external circle radius R_(2h) of the second gear;

(g) setting a contact normal g_(wD) of a drive-side tooth surface of the second gear;

(h) calculating an intersection P_(0D) between a reference plane S_(H) which is a plane orthogonal to the line of centers v_(c) and passing through the intersection C_(s) and the contact normal g_(wD) and a radius R_(20D) of the intersection P_(OD) around a gear axis;

(i) calculating an inclination angle φ_(s0D) Of a surface of action S_(wD) which is a plane defined by the pitch generating line L_(pw) and the contact normal g_(wD) with respect to the line of centers v_(c), an inclination angle ψ_(sw0D) of the contact normal g_(wD) on the surface of action S_(wD) with respect to the instantaneous axis S, and one pitch P_(gWD) on the contact normal g_(wD);

(j) setting a provisional second gear pitch cone angle Γ_(gcone), and calculating an contact ratio m_(fconeD) of the drive-side tooth surface based on the internal circle radius R_(2t) and the external circle radius R_(2h);

(k) setting a contact normal g_(wC) of a coast-side tooth surface of the second gear;

(l) calculating an intersection P_(0C) between the reference plane S_(H) which is a plane orthogonal to the line of centers v_(c) and passing through the intersection C_(s) and the contact normal g_(wC) and a radius R_(20c) of the intersection P_(0C) around the gear axis;

(m) calculating an inclination angle φ_(s0C) of a surface of action S_(wC) which is a plane defined by the pitch generating line L_(pw) and the contact normal g_(wC) with respect to the line of centers v_(c), an inclination angle ψ_(sw0C) of the contact normal g_(wC) on the surface of action S_(wC) with respect to the instantaneous axis S, and one pitch P_(gwC) on the contact normal g_(wC);

(n) setting a provisional second gear pitch cone angle Γ_(gcone), and calculating an contact ratio m_(fconeC) of the coast-side tooth surface based on the internal circle radius R_(2t) and the external circle radius R_(2h);

(o) comparing the contact ratio m_(fconeD) of the drive-side tooth surface and the contact ratio m_(fconeC) of the coast-side tooth surface, and determining whether or not these contact ratios are predetermined values;

(p) changing, when the contact ratios of the drive-side and the coast-side are not the predetermined values, the provisional second gear pitch cone angle Γ_(gcone) and re-executing from step (g);

(q) defining, when the contact ratios of the drive-side and the coast-side are the predetermined values, a virtual cone having the provisional second gear pitch cone angle Γ_(gcone) as a cone angle;

(r) calculating a provisional pitch cone angle γ_(pcone) of the virtual cone of the first gear based on the determined pitch cone angle Γ_(gcone); and

(s) calculating specifications of the hypoid gear based on the design reference point P_(w), the reference circle radius R_(1w) of the first gear and the reference circle radius R_(2w) of the second gear which are set in the step (c) and the step (d), the spiral angle ψ_(pw) of the first gear and the spiral angle ψ_(gw) of the second gear which are set in the step (c) and the step (d), the cone angle Γ_(gcone) of the virtual cone and the cone angle γ_(pcone) of the virtual cone which are defined in the step (q) and the step (r), the contact normal g_(wD) of the drive-side tooth surface of the second gear, and the contact normal g_(wC) of the coast-side tooth surface of the second gear.

According to another aspect of the present invention, in a method of designing a hypoid gear, a pitch cone angle of one gear is set equal to an inclination angle of an instantaneous axis, and the specifications are calculated. When the pitch cone angle is set equal to the inclination angle of the instantaneous axis, the contact ratios of the drive-side tooth surface and the coast-side tooth surface become almost equal to each other. Therefore, a method is provided in which the pitch cone angle is set to the inclination angle of the instantaneous axis in a simple method, that is, without reviewing the contact ratios in detail.

More specifically, according to another aspect of the present invention, the hypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of a hypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i₀, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers v_(c) with respect to rotational axes of the first gear and the second gear, an intersection C_(s) between the instantaneous axis S and the line of centers v_(c), and an inclination angle Γ_(s) of the instantaneous axis with respect to the rotational axis of the second gear;

(c) determining the inclination angle Γ_(s) of the instantaneous axis as a second gear pitch cone angle Γ_(gw); and

(d) calculating specifications of the hypoid gear based on the determined second gear pitch cone angle Γ_(gw).

According to another aspect of the present invention, in a method of designing a hypoid gear, a design reference point P_(w) is not set as a point of contact between the pitch cones of the first gear and second gear, but is determined based on one of reference circle radii R_(1w) and R_(2w) of the first gear and the second gear, a spiral angle ψ_(rw), and a phase angle β_(w) of the design reference point, and the specifications are calculated.

More specifically, according to another aspect of the present invention, a hypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of a hypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i₀, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers v_(c) with respect to rotation axes of the first gear and the second gear, an intersection C_(s) between the instantaneous axis S and the line of centers v_(c), and an inclination angle Γ_(s) of the instantaneous axis S with respect to the rotation axis of the second gear, to determine coordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting one of a reference circle radius R_(1w) of the first gear and a reference circle radius R_(2w) of the second gear, a spiral angle ψ_(rw), and a phase angle β_(w) of a design reference point P_(w), to determine the design reference point;

(d) calculating the design reference point P_(w) and a reference circle radius which is not set in the step (c) from a condition where the first gear and the second gear share the design reference point P_(w), based on the three variables which are set in the step (c);

(e) setting one of a reference cone angle γ_(pw) of the first gear and a reference cone angle Γ_(gw) of the second gear;

(f) calculating a reference cone angle which is not set in the step (e), based on the shaft angle Σ and the reference cone angle which is set in the step (e);

(g) setting a contact normal g_(wD) of a drive-side tooth surface of the second gear;

(h) setting a contact normal g_(wC) of a coast-side tooth surface of the second gear; and

(i) calculating specifications of the hypoid gear based on the design reference point P_(w), the reference circle radii R_(1w) and R_(2w), and the spiral angle ψ_(rw), which are set in the step (c) and the step (d), the reference cone angles γ_(pw) and Γ_(gw) which are set in the step (e) and the step (f), and the contact normals g_(wC) and g_(wD) which are set in the step (g) and the step (h).

The designing steps of these two aspects of the present invention can be executed by a computer by describing the steps with a predetermined computer program. A unit which receives the gear specifications and variables is connected to the computer and a unit which provides a design result or a calculation result at an intermediate stage is also connected to the computer.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a diagram showing external appearance of a hypoid gear.

FIG. 1B is a diagram showing a cross sectional shape of a gear.

FIG. 1C is a diagram showing a cross sectional shape of a pinion.

FIG. 2 is a diagram schematically showing appearance of coordinate axes in each coordinate system, a tooth surface of a gear, a tooth profile, and a path of contact.

FIG. 3 is a diagram showing a design reference point P₀ and a path of contact g₀ for explaining a variable determining method, with coordinate systems C₂′, O_(q2), C₁′, and O_(q1).

FIG. 4 is a diagram for explaining a relationship between gear axes I and II and an instantaneous axis S.

FIG. 5 is a diagram showing a relative velocity V_(s) at a point C_(s).

FIG. 6 is a diagram showing, along with planes S_(H), S_(s), S_(p), and S_(n), a reference point P₀, a relative velocity V_(rs0), and a path of contact g₀.

FIG. 7 is a diagram showing a relationship between a relative velocity V_(rs) and a path of contact g₀ at a point P.

FIG. 8 is a diagram showing a relative velocity V_(rs0) and a path of contact g₀ at a reference point P₀, with a coordinate system C_(s).

FIG. 9 is a diagram showing coordinate systems C₁, C₂, and C_(s) of a hypoid gear and a pitch generating line L_(pw).

FIG. 10 is a diagram showing a tangential cylinder of a relative velocity V_(rsw).

FIG. 11 is a diagram showing a relationship between a pitch generating line L_(pw), a path of contact g_(w), and a surface of action S_(w) at a design reference point P_(w).

FIG. 12 is a diagram showing a surface of action using coordinate systems C_(s), C₁, and C₂ in the cases of a cylindrical gear and a crossed helical gear.

FIG. 13 is a diagram showing a surface of action using coordinate systems C_(s), C₁, and C₂ in the cases of a bevel gear and a hypoid gear.

FIG. 14 is a diagram showing a relationship between a contact point P_(w) and points O_(1nw) and O_(2nw).

FIG. 15 is a diagram showing a contact point P_(w) and a path of contact g_(w) in planes S_(tw), S_(nw), and G_(2w).

FIG. 16 is a diagram showing a transmission error of a hypoid gear manufactured as prototype that uses current design method.

FIG. 17 is an explanatory diagram of a virtual pitch cone.

FIG. 18 is a diagram showing a definition of a ring gear shape.

FIG. 19 is a diagram showing a definition of a ring gear shape.

FIG. 20 is a diagram showing a state in which an addendum is extended and a tip cone angle is changed.

FIG. 21 is a diagram showing a transmission error of a hypoid gear manufactured as prototype using a design method of a preferred embodiment of the present invention.

FIG. 22 is a schematic structural diagram of a system which aids a design method for a hypoid gear.

FIG. 23 is a diagram showing a relationship between a gear ratio and a tip cone angle of a uniform tooth depth hypoid gear designed using a current method.

FIG. 24 is a diagram showing a relationship between a gear ratio and a tip cone angle of a tapered tooth depth hypoid gear designed using a current method.

FIG. 25 is a diagram showing a relationship between a tooth trace curve and a cutter radius of a uniform-depth tooth.

FIG. 26 is a diagram showing a relationship between a tooth trace curve and a cutter radius of a tapered-depth tooth.

DESCRIPTION OF THE PREFERRED EMBODIMENT

A preferred embodiment of the present invention will now be described with reference to the drawings.

1. Coordinate System of Hypoid Gear

1.1 Coordinate Systems C₁, C₂, C_(q1), and C_(q2)

In the following description, a small diameter gear in a pair of hypoid gears is referred to as a pinion, and a large diameter gear is referred to as a ring gear. In addition, in the following, the descriptions may be based on the tooth surface, tooth trace, etc. of the ring gear, but because the pinion and the ring gear are basically equivalent, the description may similarly be based on the pinion. FIG. 1A is a perspective view showing external appearance of a hypoid gear. The hypoid gear is a pair of gears in which a rotational axis (pinion axis) I of the pinion 10 and the rotational axis (gear axis) II of the ring gear 14 are not parallel and do not intersect. A line of centers v_(c) of the pinion axis and the gear axis exists, and a distance (offset) between the two axes on the line of centers v_(c) is set as E, an angle (shaft angle) between the pinion axis and the gear axis projected onto a plane orthogonal to the line of centers v_(c) is set as E, and a gear ratio is set as i₀. FIG. 1B is a cross sectional diagram at a plane including the axis II of the ring gear 14. An angle between a pitch cone element pc_(g) passing at a design reference point P_(w), to be described later, and the axis II is shown as a pitch cone angle Γ_(gw). A distance between the design reference point P_(w) and the axis II is shown as a reference circle radius R_(2w). FIG. 1C is a cross sectional diagram at a plane including the axis I of the pinion 10. An angle between a pitch cone element pc_(p) passing at the design reference point P_(w) and the axis I is shown as a pitch cone angle γ_(pw). In addition, a distance between the design reference point P_(w) and the axis I is shown as a reference circle radius R_(1w).

FIG. 2 shows coordinate systems C₁ and C₂. A direction of the line of centers v_(c) is set to a direction in which the direction of an outer product ω_(2i)×ω_(1i) of angular velocities ω_(1i) and ω_(2i) of the pinion gear axis I and the ring gear axis II is positive. The intersection points of the pinion and ring gear axes I, II and the line of centers v_(c) are designated by C₁ and C₂ and a situation where C₂ is above C₁ with respect to the line of centers v_(c) will be considered in the following. A case where C₂ is below C₁ would be very similar. A distance between C₁ and C₂ is the offset E. A coordinate system C₂ of a ring gear 14 is defined in the following manner. The origin of the coordinate system C₂ (u_(2c), v_(2c), z_(2c)) is set at C₂, a z_(2c) axis of the coordinate system C₂ is set to extend in the ω₂₀ direction on the ring gear axis II, a v_(2c) axis of the coordinate system C₂ is set in the same direction as that of the line of centers v_(c), and a u_(2c) axis of the coordinate system C₂ is set to be normal to both the axes to form a right-handed coordinate system. A coordinate system C₁ (u_(1c), v_(1c), z_(1c)) can be defined in a very similar manner for the pinion 10.

FIG. 3 shows a relationship between the coordinate systems C₁, C₂, C_(q1), and C_(q2) in gears I and II. The coordinate systems C₂ and C_(q2) of the gear II are defined in the following manner. The origin of the coordinate system C₂ (u_(2c), v_(2c), z_(2c)) is set at C₂, a z_(2c) axis of the coordinate system C₂ is set to extend in the ω₂₀ direction on the ring gear axis II, a v_(2c) axis of the coordinate system C₂ is set in the same direction as that of the line of centers v_(c), and a u_(2c) axis of the coordinate system C₂ is set to be normal to both the axes to form a right-handed coordinate system. The coordinate system C_(q2) (q_(2c), v_(q2c), z_(2c)) has the origin C₂ and the z_(2c) axis in common, and is a coordinate system formed by the rotation of the coordinate system C₂ around the z_(2c) axis as a rotational axis by χ₂₀ (the direction shown in the figure is positive) such that the plane v_(2c)=0 is parallel to the plane of action G₂₀. The u_(2c) axis becomes a q_(2c) axis, and the v_(2c) axis becomes a v_(q2c) axis.

The plane of action G₂₀ is expressed by v_(q2c)=−R_(b2) using the coordinate system C_(q2). In the coordinate system C₂, the inclination angle of the plane of action G₂₀ to the plane v_(2c)=0 is the angle χ₂₀, and the plane of action G₂₀ is a plane tangent to the base cylinder (radius R_(b20)).

The relationships between the coordinate systems C₂ and C_(q2) become as follows because the z_(2c) axis is common. u _(2c) =q _(2c) cos χ₂₀ −v _(q2c) sin χ₂₀ v _(2c) =q _(2c) sin χ₂₀ +v _(q2c) cos χ₂₀

Because the plane of action G₂₀ meets v_(q2c)=−R_(b20), the following expressions (1), are satisfied if the plane of action G₂₀ is expressed by the radius R_(b20) of the base cylinder. u _(2c) =q _(2c) cos χ₂₀ +R _(b20) sin χ₂₀ v _(2c) =q _(2c) sin χ₂₀ −R _(b20) cos χ₂₀ z _(2c) =z _(2c)  (1)

If the line of centers g₀ is defined to be on the plane of action G₂₀ and also defined such that the line of centers g₀ is directed in the direction in which the q_(2c) axis component is positive, an inclination angle of the line of centers g₀ from the q_(2c) axis can be expressed by ψ_(b20) (the direction shown in the figure is positive). Accordingly, the inclination angle of the line of centers g₀ in the coordinate system C₂ is defined to be expressed in the form of g₀ (φ₂₀, ψ_(b20)) with the inclination angle φ₂₀ (the complementary angle of the χ₂₀) of the plane of action G₂₀ with respect to the line of centers v_(c), and ψ_(b2).

As for the gear I, coordinate systems C₁ (u_(1c), v_(1c), z_(1c)) and C_(q1) (q_(1C), v_(q1c), z_(1c)), a plane of action G₁₀, a radius R_(b1) of the base cylinder, and the inclination angle g₀ (φ₁₀, ψ_(b10)) of the line of centers g₀ can be similarly defined. Because the systems share a common z_(1c) axis, the relationship between the coordinate systems C₁ and C_(q1) can also be expressed by the following expressions (2). u _(1c) =q _(1c) cos χ₁₀ +R _(b10) sin χ₁₀ v _(1c) =q _(1c) sin χ₁₀ −R _(b10) cos χ₁₀ z _(1c) =z _(1c)  (2)

The relationship between the coordinate systems C₁ and C₂ is expressed by the following expressions (3). u _(1c) =−u _(2c) cos Σ−z _(2c) sin Σ v _(1c) =v _(2c) +E z _(1c) =u _(2c) sin Σ−z _(2c) cos Σ  (3) 1.2 Instantaneous Axis (Relative Rotational Axis) S

FIG. 4 shows a relationship between an instantaneous axis and a coordinate system C_(S). If the orthogonal projections of the two axes I (ω₁₀) and II (ω₂₀) to the plane S_(H) are designated by I_(s) (ω₁₀″) and II_(s) (ω₂₀″), respectively, and an angle of I_(s) with respect to II_(S) when the plane S_(H) is viewed from the positive direction of the line of centers v_(c) to the negative direction thereof is designated by Ω, I_(s) is in a zone of 0≤Ω≤π (the positive direction of the angle Ω is the counterclockwise direction) with respect to II_(S) in accordance with the definition of ω₂₀×ω₁₀. If an angle of the instantaneous axis S (ω_(r)) to the II_(s) on the plane S_(H) is designated by Ω_(S) (the positive direction of the angle Ω_(S) is the counterclockwise direction), the components of ω₁₀″ and ω₂₀″ that are orthogonal to the instantaneous axis on the plane S_(H) must be equal to each other in accordance with the definition of the instantaneous axis (ω_(r)=ω₁₀−ω₂₀). Consequently, Ω_(s) satisfies the following expressions (4): sin Ω_(s)/sin(Ω_(s)−Ω)=ω₁₀/ω₂₀; or sin Γ_(s)/sin(Σ−Γ_(s))=ω₁₀/ω₂₀  (4) wherein Σ=π−Ω (shaft angle) and Γ_(s)=π−Ω_(s). The positive directions are shown in the figure. In other words, the angle Γ_(s) is an inclination of the instantaneous axis S with respect to the ring gear axis II_(s) on the plane S_(H), and the angle Γ_(s) will hereinafter be referred to as an inclination angle of the instantaneous axis.

The location of C_(s) on the line of centers v_(c) can be obtained as follows. FIG. 5 shows a relative velocity V_(s) (vector) of the point C_(s). In accordance with the aforesaid supposition, C₁ is located under the position of C₂ with respect to the line of centers v_(c) and ω₁₀≥ω₂₀. Consequently, C_(s) is located under C₂. If the peripheral velocities of the gears I, II at the point C_(s) are designated by V_(s1) and V_(s2) (both being vectors), respectively, because the relative velocity V_(s) (=V_(s1)−V_(s2)) exists on the instantaneous axis S, the components of V_(s1) and V_(s2) (existing on the plane S_(H)) orthogonal to the instantaneous axis must always be equal to each other. Consequently, the relative velocity V_(s) (=V_(s1)−V_(s2)) at the point C_(s) would have the shapes as shown in the same figure on the plane S_(H) according to the location (Γ_(s)) of the instantaneous axis S, and the distance C₂C_(s) between C₂ and C_(s) can be obtained by the following expression (5). That is, C ₂ C _(s) =E tan Γ_(s)/{tan(Σ−Γ_(s))+tan Γ_(s)}  (5).

The expression is effective within a range of 0≤Γ_(s)≤π, and the location of C_(s) changes together with Γ_(s), and the location of the point C_(s) is located above C₁ in the case of 0≤Γ_(s)≤π/2, and the location of the point C_(s) is located under C₁ in the case of π/2≤Γ_(s)≤π.

1.3 Coordinate System C_(s)

Because the instantaneous axis S can be determined in a static space in accordance with the aforesaid expressions (4) and (5), the coordinate system C_(s) is defined as shown in FIG. 4. The coordinate system C_(s)(u_(c), v_(c), z_(c)) is composed of C_(s) as its origin, the directed line of centers v_(c) as its v_(c) axis, the instantaneous axis S as its z_(c) axis (the positive direction thereof is the direction of ω_(r)), and its u_(c) axis taken to be normal to both the axes as a right-handed coordinate system. Because it is assumed that a pair of gears being objects transmit a motion of a constant ratio of angular velocity, the coordinate system C_(s) becomes a coordinate system fixed in the static space, and the coordinate system C_(s) is a basic coordinate system in the case of treating a pair of gears performing the transmission of the motion of constant ratio of angular velocity together with the previously defined coordinate systems C₁ and C₂.

1.4 Relationship Among Coordinate Systems C₁, C₂, and C_(s)

If the points C₁ and C₂ are expressed to be C₁ (0, v_(cs1), 0) and C₂ (0, v_(cs2), 0) by the use of the coordinate system C_(s), v_(cs1) and v_(cs2) are expressed by the following expressions (6).

$\begin{matrix} {{V_{{cs}\; 2} = {{C_{s}C_{2}} = {E\;\tan\;\Gamma_{s}\text{/}\left\{ {{\tan\left( {\Sigma - \Gamma_{s}} \right)} + {\tan\;\Gamma_{s}}} \right\}}}}\begin{matrix} {V_{{cs}\; 1} = {{C_{s}C_{1}} - V_{{cs}\; 2} - E}} \\ {= {{- E}\;{\tan\left( {\Sigma - \Gamma_{s}} \right)}\text{/}\left\{ {{\tan\left( {\Sigma - \Gamma_{s}} \right)} + {\tan\;\Gamma_{s}}} \right\}}} \end{matrix}} & (6) \end{matrix}$

If it is noted that C₂ is always located above C_(s) with respect to the v_(c) axis, the relationships among the coordinate system C_(s) and the coordinate systems C₁ and C₂ can be expressed as the following expressions (7) and (8) with the use of v_(cs1), v_(cs2), Σ, and Γ_(s). u _(1c) =u _(c) cos(Σ−Γ_(s))+z _(c) sin(Σ−Γ_(s)) v _(1c) =v _(c) −v _(cs1) z _(1c) =−u _(c) sin(Σ−Γ_(s))+z _(c) cos(Σ−Γ_(s))  (7) u _(2c) =−u _(c) cos Γ_(s) +z _(c) sin Γ_(s) v _(2c) =v _(c) −v _(cs2) z _(2c) =−u _(c) sin Γ_(s) −z _(c) cos Γ_(s)  (8)

The relationships among the coordinate system C_(s) and the coordinate systems C₁ and C₂ are conceptually shown in FIG. 6.

2. Definition of Path of Contact g₀ by Coordinate System C_(s)

2.1 Relationship Between Relative Velocity and Path of Contact g₀

FIG. 7 shows a relationship between the set path of contact g₀ and a relative velocity V_(rs) (vector) at an arbitrary point P on g₀. Incidentally, a prime sign (′) and a double-prime sign (″) in the figure indicate orthogonal projections of a point and a vector on the target plane. If the position vector of the P from an arbitrary point on the instantaneous axis S is designated by r when a tooth surface contacts at the arbitrary point P on the path of contact g₀, the relative velocity V_(rs) at the point P can be expressed by the following expression (9). v _(rs)=ω_(r) ×r+V _(s)  (9) where ω_(r)=ω₁₀−ω₂₀ ω_(r)=ω₂₀ sin Σ/sin(Σ−Γ_(s))=ω₁₀ sin Σ/sin Γ_(s) V _(s)=ω₁₀×[C ₁ C _(s)]−ω₂₀×[C ₂ C _(s)] V _(s)=ω₂₀ E sin Γ_(s)=ω₁₀ E sin(Σ−Γ_(s))

Here, [C₁C_(s)] indicates a vector having C₁ as its starting point and C_(s) as its end point, and [C₂C_(s)] indicates a vector having C₂ as its starting point and C_(s) as its end point.

The relative velocity V_(rs) exists on a tangential plane of the surface of a cylinder having the instantaneous axis S as an axis, and an inclination angle ψ relative to V_(s) on the tangential plane can be expressed by the following expression (10). cos ψ=|V _(s) |/|V _(rs)|  (10)

Because the path of contact g₀ is also the line of centers of a tooth surface at the point of contact, g₀ is orthogonal to the relative velocity V_(rs) at the point P. That is, V _(rs) ·g ₀=0

Consequently, g₀ is a directed straight line on a plane N normal to V_(rs) at the point P. If the line of intersection of the plane N and the plane S_(H) is designated by H_(n), H_(n) is in general a straight line intersecting with the instantaneous axis S, with g₀ necessarily passing through the H_(n) if an infinite intersection point is included. If the intersection point of g₀ with the plane S_(H) is designated by P₀, then P₀ is located on the line of intersection H_(n), and g₀ and P₀ become as follows according to the kinds of pairs of gears.

(1) Case of Cylindrical Gears or Bevel Gears (Σ=0, π or E=0)

Because V_(s)=0, V_(rs) simply means a peripheral velocity around the instantaneous axis S. Consequently, the plane N includes the S axis. Hence, H_(n) coincides with S, and the path of contact g₀ always passes through the instantaneous axis S. That is, the point P₀ is located on the instantaneous axis S. Consequently, for these pairs of gears, the path of contact g₀ is an arbitrary directed straight line passing at the arbitrary point P₀ on the instantaneous axis.

(2) Case of Gear Other than that Described Above (Σ≠0, π or E≠0)

In the case of a hypoid gear, a crossed helical gear or a worm gear, if the point of contact P is selected at a certain position, the relative velocity V_(rs), the plane N, and the straight line H_(n), all peculiar to the point P, are determined. The path of contact g₀ is a straight line passing at the arbitrary point P₀ on H_(n), and does not, in general, pass through the instantaneous axis S. Because the point P is arbitrary, g₀ is also an arbitrary directed straight line passing at the point P₀ on a plane normal to the relative velocity V_(rs0) at the intersection point P₀ with the plane S_(H). That is, the aforesaid expression (9) can be expressed as follows. V _(rs) =V _(rs0)+ω_(r)×[P ₀ P]·g ₀

Here, [P₀P] indicates a vector having P₀ as its starting point and the P as its end point. Consequently, if V_(rs0)·g₀=0, V_(rs)·g₀=0, and the arbitrary point P on g₀ is a point of contact.

2.2 Selection of Reference Point

Among pairs of gears having two axes with known positional relationship and the angular velocities, pairs of gears with an identical path of contact g₀ have an identical tooth profile corresponding to g₀, with the only difference between them being which part of the tooth profile issued effectively. Consequently, in design of a pair of gears, the position at which the path of contact g₀ is disposed in a static space determined by the two axes is important. Further, because a design reference point is only a point for defining the path of contact g₀ in the static space, the position at which the design reference point is selected on the path of contact g₀ does not cause any essential difference. When an arbitrary path of contact g₀ is set, the g₀ necessarily intersects with a plane S_(H) including the case where the intersection point is located at an infinite point. Thus, the path of contact g₀ is determined with the point P₀ on the plane S_(H) (on an instantaneous axis in the case of cylindrical gears and bevel gears) as the reference point.

FIG. 8 shows the reference point P₀ and the path of contact g₀ by the use of the coordinate system C_(s). When the reference point expressed by means of the coordinate system C_(s) is designated by P₀ (u_(c0), v_(c0), z_(c0)), each coordinate value can be expressed as follows. u _(c0) =O _(s) P ₀ V _(c0)=0 z _(c0) =C _(s) O _(s)

For cylindrical gears and bevel gears, u_(c0)=0. Furthermore, the point O_(s) is the intersection point of a plane S_(s), passing at the reference point P₀ and being normal to the instantaneous axis S, and the instantaneous axis S.

2.3 Definition of Inclination Angle of Path of Contact g₀

The relative velocity V_(rs0) at the point P₀ is concluded as follows with the use of the aforesaid expression (9). V _(rs0)=ω_(r)×[u _(c0)]+V _(s) where, [u_(c0)] indicates a vector having O_(s) as its starting point and P₀ as its end point. If a plane (u_(c)=u_(c0)) being parallel to the instantaneous axis S and being normal to the plane S_(H) at the point P₀ is designated by S_(p), v_(rs0) is located on the plane S_(p), and the inclination angle ψ₀ of V_(rs0) from the plane S_(H) (v_(c)=0) can be expressed by the following expression (11) with the use of the aforesaid expression (10).

$\begin{matrix} \begin{matrix} {{\tan\;\psi_{0}} = {\omega_{r}u_{c\; 0}\text{/}V_{s}}} \\ {= {u_{c\; 0}\sin\;\Sigma\text{/}\left\{ {E\;{\sin\left( {\Sigma - \Gamma_{s}} \right)}\sin\;\Gamma_{s}} \right\}}} \end{matrix} & (11) \end{matrix}$

Incidentally, ψ₀ is assumed to be positive when u_(c0)≥0, and the direction thereof is shown in FIG. 8.

If a plane passing at the point P₀ and being normal to V_(rs0) is designated by S_(n), the plane S_(n) is a plane inclining to the plane S_(s) by the ψ₀, and the path of contact g₀ is an arbitrary directed straight line passing at the point P₀ and located on the plane S_(n). Consequently, the inclination angle of g₀ in the coordinate system C_(s) can be defined with the inclination angle ψ₀ of the plane S_(n) from the plane S_(s) (or the v_(c) axis) and the inclination angle φ_(n0) from the plane S_(p) on the plane S_(n), and the defined inclination angle is designated by g₀ (ψ₀, φ_(n0)). The positive direction of φ_(n0) is the direction shown in FIG. 8.

2.4. Definition of g₀ by Coordinate System C_(s)

FIG. 6 shows relationships among the coordinate system C_(s), the planes S_(H), S_(s), S_(p) and S_(n), P₀ and g₀(ψ₀, φ_(n0)). The plane S_(H) defined here corresponds to a pitch plane in the case of cylindrical gears and an axial plane in the case of a bevel gear according to the current theory. The plane S_(s) is a transverse plane, and the plane S_(p) corresponds to the axial plane of the cylindrical gears and the pitch plane of the bevel gear. Furthermore, it can be considered that the plane S_(n) is a normal plane expanded to a general gear, and that φ_(n0) and ψ₀ also are a normal pressure angle and a spiral angle expanded to a general gear, respectively. By means of these planes, pressure angles and spiral angles of a pair of general gears can be expressed uniformly to static spaces as inclination angles to each plane of line of centers (g₀'s in this case) of points of contact. The planes S_(n), φ_(n0), and φ₀ defined here coincide with those of a bevel gear of the current theory, and differ for other gears because the current theory takes pitch planes of individual gears as standards, and then the standards change to a static space according to the kinds of gears. With the current theory, if a pitch body of revolution (a cylinder or a circular cone) is determined, it is sufficient to generate a mating surface by fixing an arbitrary curved surface to the pitch body of revolution as a tooth surface, and in the current theory, conditions of the tooth surface (a path of contact and the normal thereof) are not limited except for the limitations of manufacturing. Consequently, the current theory emphasizes the selection of P₀ (for discussions about pitch body of revolution), and there has been little discussion concerning design of g₀ (i.e. a tooth surface realizing the g₀) beyond the existence of a tooth surface.

For a pair of gears having the set shaft angle Σ thereof, the offset E thereof, and the directions of angular velocities, the path of contact g₀ can generally be defined in the coordinate system C_(s) by means of five independent variables of the design reference point P₀ (u_(c0), v_(c0), z_(c0)) and the inclination angle g₀ (ψ₀, φ_(n0)). Because the ratio of angular velocity i₀ and v_(c0)=0 are set as design conditions in the present embodiment, there are three independent variables of the path of contact g₀. That is, the path of contact g₀ is determined in a static space by the selections of the independent variables of two of (z_(c0)), φ_(n0), and ψ₀ in the case of cylindrical gears because z_(c0) has no substantial meaning, three of z_(c0), φ_(n0), and ψ₀ in the case of a bevel gear, or three of z_(c0), φ_(n0), and ψ₀ (or u_(c0)) in the case of a hypoid gear, a worm gear, or a crossed helical gear. When the point P₀ is set, ψ₀ is determined at the same time and only φ_(n0) is a freely selectable variable in the case of the hypoid gear and the worm gear. However, in the case of the cylindrical gears and the bevel gear, because P₀ is selected on an instantaneous axis, both of ψ₀ and φ_(n0) are freely selectable variables.

3. Pitch Hyperboloid

3.1 Tangential Cylinder of Relative Velocity

FIG. 9 is a diagram showing an arbitrary point of contact P_(w), a contact normal g_(w) thereof, a pitch plane S_(tw), the relative velocity V_(rsw), and a plane S_(nw) which is normal to the relative velocity V_(rsw) of a hypoid gear, along with basic coordinate systems C₁, C₂, and C_(s). FIG. 10 is a diagram showing FIG. 9 drawn from a positive direction of the z_(c) axis of the coordinate system C_(s). The arbitrary point P_(w) and the relative velocity V_(rsw) are shown with cylindrical coordinates P_(w)(r_(w), β_(w), z_(cw): C_(s)). The relative velocity V_(rsw) is inclined by ψ_(rw) from a generating line L_(pw) on the tangential plane S_(pw) of the cylinder having the z_(c) axis as its axis, passing through the arbitrary point P_(w), and having a radius of r_(w).

When the coordinate system C_(s) is rotated around the z_(c) axis by β_(w), to realize a coordinate system C_(rs) (u_(rc), v_(rc), z_(c): C_(rs)), the tangential plane S_(pw) can be expressed by u_(rc)=r_(w), and the following relationship is satisfied between u_(rc)=r_(w) and the inclination angle ψ_(rw) of V_(rsw).

$\begin{matrix} \begin{matrix} {u_{rc} = {r_{w} = {V_{s}\tan\;\psi_{rw}\text{/}\omega_{r}}}} \\ {= {E\;\tan\;\psi_{rw} \times {\sin\left( {\Sigma - \Gamma_{s}} \right)}\sin\;\Gamma_{s}\text{/}\sin\;\Sigma}} \end{matrix} & (12) \end{matrix}$ where Vs represents a sliding velocity in the direction of the instantaneous axis and ω_(r) represents a relative angular velocity around the instantaneous axis.

The expression (12) shows a relationship between r_(w) of the arbitrary point P_(w) (r_(w), β_(w), z_(cw): C_(s)) and the inclination angle ψ_(rw) of the relative velocity V_(rsw) thereof. In other words, when ψ_(rw) is set, r_(w) is determined. Because this is true for arbitrary values of β_(w), and z_(cw), P_(w) with a constant ψ_(rw) defines a cylinder with a radius r_(w). This cylinder is called the tangential cylinder of the relative velocity.

3.2 Pitch Generating Line and Surface of Action

When r_(w) (or ψ_(pw)) and β_(w) are set, P_(w) is determined on the plane z_(c)=z_(cw). Because this is true for an arbitrary value of z_(cw), points P_(w) having the same r_(w) (or ψ_(rw)) and the same β_(w) draw a line element of the cylinder having a radius r_(w). This line element is called a pitch generating line L_(pw). A directed straight line which passes through a point P_(w) on a plane S_(nw) orthogonal to the relative velocity V_(rsw) at the arbitrary point P_(w) on the pitch generating line L_(pw) satisfies a condition of contact, and thus becomes a contact normal.

FIG. 11 is a diagram conceptually drawing relationships among the pitch generating line L_(pw), directed straight line g_(w), surface of action S_(w), contact line w, and a surface of action S_(wc) (dotted line) on the side C to be coast. A plane having an arbitrary directed straight line g_(w) on the plane S_(nw) passing through the point P_(w) as a normal is set as a tooth surface W. Because all of the relative velocity V_(rsw) at the arbitrary point P_(w) on the pitch generating line L_(pw) are parallel and the orthogonal planes S_(nw) are also parallel, of the normals of the tooth surface W, any normal passing through the pitch generating line L_(pw) becomes a contact normal, and a plane defined by the pitch generating line L_(pw) and the contact normal g_(w) becomes the surface of action S_(w) and an orthogonal projection of the pitch generating line L_(pw) to the tooth surface W becomes the contact line w. Moreover, because the relationship is similarly true for another normal g_(wc) on the plane S_(nw) passing through the point P_(w) and the surface of action S_(wc) thereof, the pitch generating line L_(pw) is a line of intersection between the surfaces of action of two tooth surfaces (on the drive-side D and coast-side C) having different contact normals on the plane S_(nw).

3.3 Pitch Hyperboloid

The pitch generating line L_(pw) is uniquely determined by the shaft angle Σ, offset E, gear ratio i₀, inclination angle ψ_(rw) of relative velocity V_(rsw), and rotation angle β_(w) from the coordinate system C_(s) to the coordinate system C_(rs). A pair of hyperboloids which are obtained by rotating the pitch generating line L_(pw) around the two gear axes, respectively, contact each other in a line along L_(pw), and because the line L_(pw) is also a line of intersection between the surfaces of action, the drive-side D and the coast-side C also contact each other along the line L_(pw). Therefore, the hyperboloids are suited as revolution bodies for determining the outer shape of the pair of gears. In the present invention, the hyperboloids are set as the design reference revolution bodies, and are called the pitch hyperboloids. The hyperboloids in the related art are revolution bodies in which the instantaneous axis S is rotated around the two gear axes, respectively, but in the present invention, the pitch hyperboloid is a revolution body obtained by rotating a parallel line having a distance r_(w) from the instantaneous axis.

In the cylindrical gear and the bevel gear, L_(pw) coincides with the instantaneous axis S or z_(c) (r_(w)→0) regardless of ψ_(rw) and β_(w), because of special cases of the pitch generating line L_(pw) (V_(s)→0 as Σ→0 or E→0 in the expression (12)). The instantaneous axis S is a line of intersection of the surfaces of action of the cylindrical gear and the bevel gear, and the revolution bodies around the gear axes are the pitch cylinder of the cylindrical gear and the pitch cone of the bevel gear.

For these reasons, the pitch hyperboloids which are the revolution bodies of the pitch generating line L_(pw) have the common definition of the expression (12) from the viewpoint that the hyperboloid is a “revolution body of line of intersection of surfaces of action” and can be considered to be a design reference revolution body for determining the outer shape of the pair of gears which are common to all pairs of gears.

3.4 Tooth Trace (New Definition of Tooth Trace)

In the present invention, a curve on the pitch hyperboloid (which is common to all gears) obtained by transforming a path of contact to a coordinate system which rotates with the gear when the tooth surface around the point of contact is approximated with its tangential plane and the path of contact is made coincident with the line of intersection of the surfaces of action (pitch generating line L_(pw)) is called a tooth trace (curve). In other words, a tooth profile, among arbitrary tooth profiles on the tooth surface, in which the path of contact coincides with the line of intersection of the surface of action is called a tooth trace. The tooth trace of this new definition coincides with the tooth trace of the related art defined as an intersection between the pitch surface (cone or cylinder) and the tooth surface in the cylindrical gears and the bevel gears and differs in other gears. In the case of the current hypoid gear, the line of intersection between the selected pitch cone and the tooth surface is called a tooth trace.

3.5 Contact Ratio

A total contact ratio m is defined as a ratio of a maximum angular displacement and an angular pitch of a contact line which moves on an effective surface of action (or effective tooth surface) with the rotation of the pair of gears. The total contact ratio m can be expressed as follows in terms of the angular displacement of the gear. m=(θ_(2max)−θ_(2min))/(2θ_(2p)) where θ_(2max) and θ_(2min) represent maximum and minimum gear angular displacements of the contact line and 2θ_(2p) represents a gear angular pitch.

Because it is very difficult to represent the position of the contact line as a function of a rotation angle except for special cases (involute helicoid) and it is also difficult to represent such on the tooth surface (curved surface), in the stage of design, the surface of action has been approximated with a plane in a static space, a path of contact has been set on the surface of action, and an contact ratio has been determined and set as an index along the path of contact.

FIGS. 12 and 13 show the surface of action conceptually shown in FIG. 11 in more detail with reference to the coordinate systems C_(s), C₁, and C₂. FIG. 12 shows a surface of action in the cases of the cylindrical gear and a crossed helical gear, and FIG. 13 shows a surface of action in the cases of the bevel gear and the hypoid gear. FIGS. 12 and 13 show surfaces of action with the tooth surface (tangential plane) when the intersection between g_(w) and the reference plane S_(H) (v_(c)=0) is set as P₀ (u_(c0), v_(c0)=0, z_(c0): C_(s)), the inclination angle of g_(w) is represented in the coordinate system C_(s), and the contact normal g_(w) is set g_(w)=g₀ (ψ₀, φ_(n0): C_(s)). A tooth surface passing through P₀ is shown as W₀, a tooth surface passing through an arbitrary point P_(d) on g_(w)=g₀ is shown as W_(d), a surface of action is shown by S_(w)=S_(w0), and an intersection between the surface of action and the plane S_(H) is shown with L_(pw0) (which is parallel to L_(pw)). Because planes are considered as the surface of action and the tooth surface, the tooth surface translates on the surface of action. The point P_(w) may be set at any point, but because the static coordinate system C_(s) has its reference at the point P₀ on the plane S_(H), the contact ratio is defined with an example configuration in which P_(w) is set at P₀.

The contact ratio of the tooth surface is defined in the following manner depending on how the path of contact passing through P_(w)=P₀ is defined on the surface of action S_(w)=S_(w0):

(1) Contact Ratio m_(z) Orthogonal Axis

This is a ratio between a length separated by an effective surface of action (action limit and the tooth surface boundary) of lines of intersection h_(1z) and h_(2z) (P₀P_(z1sw) and P₀P_(z2sw) in FIGS. 12 and 13) between the surface of action S_(w0) and the planes of rotation Z₁₀ and Z₂₀ and a pitch in this direction;

(2) Tooth Trace Contact Ratio m_(f)

This is a ratio between a length of L_(pw0) which is parallel to the instantaneous axis separated by the effective surface of action and a pitch in this direction;

(3) Transverse Contact Ratio m_(s)

This is a ratio between a length separated by an effective surface of action of a line of intersection (P₀P_(ssw) in FIGS. 12 and 13) between a plane S_(s) passing through P₀ and normal to the instantaneous axis and S_(w0) and a pitch in this direction;

(4) Contact Ratio in Arbitrary Direction

This includes cases where the path of contact is set in a direction of g₀ (P₀P_(Gswn) in FIG. 13) and cases where the path of contact is set in a direction of a line of intersection (P_(w)P_(gcon) in FIGS. 12 and 13) between an arbitrary conical surface and S_(w0);

(5) Total Contact Ratio

This is a sum of contact ratios in two directions (for example, (2) and (3)) which are normal to each other on the surface of action, and is used as a substitute for the total contact ratio.

In addition, except for points on g_(w)=g₀, the pitch (length) would differ depending on the position of the point, and the surface of action and the tooth surface are actually not planes. Therefore, only an approximated value can be calculated for the contact ratio. Ultimately, a total contact ratio determined from the angular displacement must be checked.

3.6 General Design Method of Gear Using Pitch Hyperboloid

In general, a gear design can be considered, in a simple sense, to be an operation, in a static space (coordinate system C_(s)) determined by setting the shaft angle Σ, offset E, and gear ratio i₀, to:

(1) select a pitch generating line and a design reference revolution body (pitch hyperboloid) by setting a design reference point P_(w)(r_(w)(ψ_(rw)), β_(w), z_(cw): C_(rs)); and

(2) set a surface of action (tooth surface) having g_(w) by setting an inclination angle (ψ_(rw), φ_(nrw): C_(rs)) of a tooth surface normal g_(w) passing through P_(w).

In other words, the gear design method (selection of P_(w) and g_(w)) comes down to selection of four variables including r_(w) (normally, ψ_(rw), is set), β_(w), z_(cw) (normally, R_(2w) (gear pitch circle radius) is set in place of z_(cw)), and φ_(nrw). A design method for a hypoid gear based on the pitch hyperboloid when Σ, E, and i₀ are set will be described below.

3.7 Hypoid Gear (−π/2<β_(w)<π/2)

(1) Various hypoid gears can be realized depending on how β_(w) is selected, even with set values for φ_(rw) (r_(w)) and z_(cw) (R_(2w)).

(a) From the viewpoint of the present invention, the Wildhaber (Gleason) method is one method of determining P_(w) by determining β_(w) through setting of a constraint condition to “make the radius of curvature of a tooth trace on a plane (FIG. 9) defined by peripheral velocities of a pinion and ring gear at P_(w) coincide with the cutter radius”. However, because the tooth surface is possible as long as an arbitrary curved surface (therefore, arbitrary radius of curvature of tooth trace) having g_(w) passing through P_(w) has a mating tooth surface, this condition is not necessarily a requirement even when a conical cutter is used. In addition, although this method employs circular cones which circumscribe at P_(w), the pair of gears still contact on a surface of action having the pitch generating line L_(pw) passing through P_(w) regardless of the cones. Therefore, the line of intersection between the pitch cone circumscribing at P_(w) and the surface of action determined by this method differs from the pitch generating line L_(pw) (line of intersection of surfaces of action). When g_(w) is the same, the inclination angle of the contact line on the surface of action and L_(pw) are equal to each other, and thus the pitch in the direction of the line of intersection between the surface of action and the pitch cone changes according to the selected pitch cone (FIG. 11). In other words, a large difference in the pitch is caused between the drive-side and the coast-side in the direction of the line of intersection between the pitch cone and the surface of action (and, consequently, the contact ratios in this direction). In the actual Wildhaber (Gleason) method, two cones are determined by giving pinion spiral angle and an equation of radius of curvature of tooth trace for contact equations of the two cones (seven equations having nine unknown variables), and thus the existence of the pitch generating line and the pitch hyperboloid is not considered.

(b) In a preferred embodiment described in section 4.2A below, β_(w) is selected by giving a constraint condition that “a line of intersection between a cone circumscribing at P_(w) and the surface of action is coincident with the pitch generating line L_(pw)”. As a result, as will be described below, the tooth trace contact ratios on the drive-side and the coast-side become approximately equal to each other.

(2) Gear radius R_(2w), β_(w), and ψ_(rw) are set and a design reference point P_(w)(u_(cw), v_(cw), z_(cw): C_(s)) is determined on the pitch generating line L_(pw). The pitch hyperboloids can be determined by rotating the pitch generating line L_(pw) around each tooth axis. A method of determining the design reference point will be described in section 4.2B below.

(3) A tooth surface normal g_(w) passing through P_(w) is set on a plane S_(nw) normal to the relative velocity V_(rsw) of P_(w). The surface of action S_(w) is determined by g_(w) and the pitch generating line L_(pw).

4. Design Method for Hypoid Gear

A method of designing a hypoid gear using the pitch hyperboloid will now be described in detail.

4.1 Coordinate Systems C_(s), C₁, and C₂ and Reference Point P_(w)

When the shaft angle Σ, offset E, and gear ratio i₀ are set, the inclination angle Γ_(s) of the instantaneous axis, and the origins C₁(0, V_(cs1), 0: C_(s)) and C₂(0, v_(cs2), 0: C_(s)) of the coordinate systems C₁ and C₂ are represented by the following expressions. sin Γ_(s)/sin(Σ−Γ_(s))=i ₀ v _(cs2) =E tan Γ_(s)/{tan(Σ−Γ_(s))+tan Γ_(s)} v _(cs1) =v _(cs2) −E

The reference point P_(w) is set in the coordinate system C_(s) as follows. P _(w)(u _(cw) ,v _(cw) ,z _(cw) :C _(s))

If P_(w) is set as P_(w)(r_(w), β_(w), z_(cw): C_(s)) by representing P_(w) with the cylindrical radius r_(w) of the relative velocity and the angle β_(w) from the u_(c) axis, the following expressions hold. u _(cw) =r _(w) cos β_(w) v _(cw) =r _(w) sin β_(w)

The pitch generating line L_(pw) is determined as a straight line which passes through the reference point P_(w) and which is parallel to the instantaneous axis (inclination angle Γ_(s)), and the pitch hyperboloids are determined as revolution bodies of the pitch generating line L_(pw) around the gear axes.

If the relative velocity of P_(w) is V_(rsw), the angle ψ_(rw) between V_(rsw) and the pitch generating line L_(pw) is, based on expression (12), tan ψ_(rw) =r _(w) sin Σ/{E sin(Σ−Γ_(s))sin Γ_(s)}

Here, ψ_(rw) is the same anywhere on the same cylinder of the radius r_(w).

When transformed into coordinate systems C₁ and C₂, P_(w)(u_(1cw), v_(1cw), z_(1cw): C₁), P_(w)(u_(2cw), v_(2cw), z_(2cw): C₂), and pinion and ring gear reference circle radii R_(1w) and R_(2w) can be expressed with the following expressions. u _(1cw) =u _(cw) cos(Σ−Γ_(s))+z _(cw) sin(Σ−Γ_(s)) v _(1cw) =v _(cw) −v _(cs1) z _(1cw) =−u _(w) sin(Σ−Γ_(s))+z _(cw) cos(Σ−Γs) u _(2cw) =−u _(cw) cos Γ_(s) +z _(cw) sin Γ_(s) v _(2cw) =v _(cw) −v _(cs2) z _(2cw) =−u _(cw) sin Γ_(s) −z _(cw) cos Γ_(s) R _(1w) ² =u _(1cw) ² +v _(1cw) ² R _(2w) ² =u _(2cw) ² +v _(2cw) ²  (13) 4.2A Cones Passing Through Reference Point P_(w)

A pitch hyperboloid which is a geometric design reference revolution body is difficult to manufacture, and thus in reality, in general, the gear is designed and manufactured by replacing the pitch hyperboloid with a pitch cone which passes through the point of contact P_(w). The replacement with the pitch cones is realized in the present embodiment by replacing with cones which contact at the point of contact P_(w).

The design reference cone does not need to be in contact at P_(w), but currently, this method is generally practiced. When β_(w) is changed, the pitch angle of the cone which contacts at P_(w) changes in various manners, and therefore another constraint condition is added for selection of the design reference cone (β_(w)) The design method would differ depending on the selection of the constraint condition. One of the constraint conditions is the radius of curvature of the tooth trace in the Wildhaber (Gleason) method which is already described. In the present embodiment, β_(w) is selected with a constraint condition that a line of intersection between the cone which contacts at P_(w) and the surface of action coincides with the pitch generating line L_(pw).

As described, there is no substantial difference caused by where on the path of contact g₀ the design reference point is selected. Therefore, a design method of a hypoid gear will be described in which the point of contact P_(w) is set as the design reference point and circular cones which contact at P_(w) are set as the pitch cones.

4.2A.1 Pitch Cone Angles

Intersection points between a plane S_(nw) normal to the relative velocity V_(rsw) of the reference point P_(w) and the gear axes are set as O_(1nw) and O_(2nw) (FIG. 9). FIG. 14 is a diagram showing FIG. 9 viewed from the positive directions of the tooth axes z_(1c) and z_(2c), and intersections O_(1nw) and O_(2nw) can be expressed by the following expressions. O _(1nw)(0,0,−E/(tan ε_(2w) sin Σ):C ₁) O _(2nw)(0,0,−E/(tan ε_(1w) sin Σ):C ₂) where sin ε_(1w)=v_(1cw)/R_(1w) and sin ε_(2w)=v_(2cw)/R_(2w).

In addition, O_(1nw)P_(w) and O_(2nw)P_(w) can be expressed with the following expressions. O _(1nw) P _(w) ={R _(1w) ²+(−E/(tan ε_(2w) sin Σ)−z _(1cw))²}^(1/2) O _(2nw) P _(w) ={R _(2w) ²+(−E/(tan ε_(1w) sin Σ)−z _(2cw))²}^(1/2)

Therefore, the cone angles γ_(pw) and Γ_(gw) of the pinion and ring gear can be determined with the following expressions, taking advantage of the fact that O_(1nw)P_(w) and O_(2nw)P_(w) are back cone elements: cos γ_(pw) =R _(1w) /O _(1nw) P _(w) cos Γ_(gw) =R _(2w) /O _(2nw) P _(w)  (14)

The expression (14) sets the pitch cone angles of cones having radii of R_(1w) and R_(2w) and contacting at P_(w).

4.2A.2 Inclination Angle of Relative Velocity at Reference Point P_(w)

The relative velocity and peripheral velocity areas follows. V _(rsw)/ω₂₀={(E sin Γ_(s))²+(r _(w) sin Σ/sin(Σ−Γ_(s)))²}^(1/2) V _(1w)/ω₂₀ =i ₀ R _(1w) V _(2w)/ω₂₀ =R _(2w)

When a plane defined by peripheral velocities V_(1w) and V_(2w) is S_(tw), the plane S_(tw) is a pitch plane. If an angle formed by V_(1w) and V_(2w) is ψ_(v12w) and an angle formed by V_(rsw) and V_(1w) is ψ_(vrs1w) (FIG. 9), cos(ψ_(v12w))=(V _(1w) ² +V _(2w) ² −V _(rsw) ²)/(2V _(1w) ×V _(2w)) cos(ψ_(vrs1w))=(V _(rsw) ² +V _(1w) ² −V _(2w) ²)/(2V _(1w) ×V _(rsw))

If the intersections between the plane S_(tw) and the pinion and gear axes are O_(1w) and O_(2w), the spiral angles of the pinion and the ring gear can be determined in the following manner as inclination angles on the plane S_(tw) from P_(w)O_(1w) and P_(w)O_(2w) (FIG. 9). ψ_(pw)=π/2−ψ_(vrs1w) ψ_(gw)=π/2−ψ_(v12w)−ψ_(vrs1w)  (15)

When a pitch point P_(w)(r_(w), β_(w), z_(cw): C_(s)) is set, specifications of the cones contacting at P_(w) and the inclination angle of the relative velocity V_(rsw) can be determined based on expressions (13), (14) and (15). Therefore, conversely, the pitch point P_(w) and the relative velocity V_(rsw) can be determined by setting three variables (for example, R_(2w), ψ_(pw), Γ_(gw)) from among the cone specifications and the inclination angle of the relative velocity V_(rsw). Each of these three variables may be any variable as long as the variable represents P_(w), and the variables may be, in addition to those described above, for example, a combination of a ring gear reference radius R_(2w)r a ring gear spiral angle ψ_(gw), and a gear pitch cone angle Γ_(gw), or a combination of the pinion reference radius R_(1w), the ring gear spiral angle ψ_(pw), and Γ_(gw).

4.2A.3 Tip Cone Angle

Normally, an addendum a_(G) and an addendum angle α_(G)=a_(G)/O_(2w)P_(w) are determined and the tip cone angle is determined by Γ_(gf)=Γ_(s)+α_(G). Alternatively, another value may be arbitrarily chosen for the addendum angle α_(G).

4.2A.4 Inclination Angle of Normal g_(w) at Reference Point P_(w)

FIG. 15 shows the design reference point P_(w) and the contact normal g_(w) on planes S_(tw), S_(nw), and G_(2w).

(1) Expression of Inclination Angle of g_(w) in Coordinate System C_(s)

An intersection between g_(w) passing through P_(w)(u_(cw), v_(cw), z_(cw): C_(s)) and the plane S_(H) (β_(w)=0) is set as P₀(u_(c0), 0, z_(c0): C_(s)) and the inclination angle of g_(w) is represented with reference to the point P₀ in the coordinate system C_(s), by g_(w) (ψ₀, φ_(n0): C_(s)). The relationship between P₀ and P_(w) is as follows (FIG. 11): u _(c0) =u _(cw)+(v _(cw)/cos ψ₀)tan φ_(n0) z _(c0) =z _(cw) −v _(cw) tan ψ₀  (16) (2) Expression of Inclination Angle of g_(w) on Pitch Plane S_(tw) and Plane S_(nw) (FIG. 9)

When a line of intersection between the plane S_(nw) and the pitch plane S_(tw) is g_(tw), an inclination angle on the plane S_(nW) from g_(tw) is set as φ_(nw). The inclination angle of g_(w) is represented by g_(w)(ψ_(gw), φ_(nw)) using the inclination angle ψ_(gw) of V_(rsw) from P_(w)O_(2w) on the pitch plane S_(tw) and φ_(nw).

(3) Transformation Equation of Contact Normal g_(w)

In the following, transformation equations from g_(w)(ψ_(gw), φ_(nw)) to g_(w)(ψ₀, φ_(n0): C_(s)) will be determined.

FIG. 15 shows g_(w)(ψ_(gw), φ_(nw)) with g_(w)(φ_(2w), ψ_(b2w): C₂). In FIG. 15, g_(w) is set with P_(w)A, and projections of point A are sequentially shown with B, C, D, and E. In addition, the projection points to the target sections are shown with prime signs (′) and double-prime signs (″). The lengths of the directed line segments are determined in the following manner, with P_(w)A=L_(g):

$\begin{matrix} {\mspace{85mu}{{{{A^{\prime}A} = {L_{g}\sin\;\varphi_{nw}}}\mspace{20mu}{{B^{\prime}B} = {L_{g}\cos\;\varphi_{nw}\cos\;\psi_{gw}}}\mspace{20mu}{{C^{\prime}C} = {A^{\prime}A}}\mspace{20mu}{{P_{w}C^{\prime}} = {L_{g}\cos\;\varphi_{nw}\sin\;\psi_{gw}}}\mspace{20mu}{{C^{\prime}K} = {P_{w}C^{\prime}\text{/}\tan\;\Gamma_{gw}}}}\begin{matrix} {\left. \mspace{79mu}{{C^{''}C} = {{C^{\prime}C} - {C^{\prime}K}}} \right)\sin\;\Gamma_{gw}} \\ {{L_{g}\left( {{\sin\;\varphi_{nw}} - {\cos\;\varphi_{nw}\sin\;\psi_{gw}\text{/}\tan\;\Gamma_{gw}}} \right)}\sin\;\Gamma_{gw}} \end{matrix}\mspace{20mu}{{D^{\prime}D} = {B^{\prime}B}}\mspace{20mu}{{P_{w}E} = {P_{w}A}}\mspace{79mu}{{E^{\prime}E} = {C^{''}C}}\begin{matrix} {\mspace{79mu}{{\sin\;\psi_{b\; 2w}} = {{E^{\prime}E\text{/}P_{w}E} = {C^{''}C\text{/}L_{g}}}}} \\ {= {\left( {{\sin\;\varphi_{nw}} - {\cos\;\varphi_{nw}\sin\;\psi_{gw}\text{/}\tan\;\Gamma_{gw}}} \right)\sin\;\Gamma_{gw}}} \end{matrix}}} & (17) \\ {\mspace{79mu}{{{\tan\;\eta_{x\; 2w}} = {{C^{\prime}C\text{/}P_{w}C^{\prime}} = {\tan\;\varphi_{nw}\text{/}\sin\;\psi_{gw}}}}{{P_{w}C} = {\left( {{P_{w}C^{\prime\; 2}} + {C^{\prime}C^{2}}} \right)^{1/2} = {L_{g} \times \left\{ {\left( {\cos\;\varphi_{nw}\sin\;\psi_{gw}} \right)^{2} + \left( {\sin\;\varphi_{nw}} \right)^{2}} \right\}^{1/2}}}}\mspace{79mu}{{P_{w}C^{''}} = {{P_{w}C\;\cos\left\{ {\eta_{x\; 2w} - \left( {{\pi\text{/}2} - \Gamma_{gw}} \right)} \right\}} = {P_{w}C\;{\sin\left( {\eta_{{xw}\; 2} + \Gamma_{gw}} \right)}}}}\begin{matrix} {\mspace{79mu}{{\tan\left( {\chi_{2w} - ɛ_{2w}} \right)} = {D^{\prime}D\text{/}P_{w}C^{''}}}} \\ {= {\cos\;\varphi_{nw}\cos\;\psi_{gw}{\text{/}\left\lbrack \left\{ {\left( {\cos\;\varphi_{nw}\sin\;\psi_{gw}} \right)^{2} +} \right. \right.}}} \\ \left. {\left. \left( {\sin\;\varphi_{nw}} \right)^{2} \right\}^{1/2} \times {\sin\left( {\eta_{x\; 2w} + \Gamma_{gw}} \right)}} \right\rbrack \end{matrix}\mspace{20mu}{\varphi_{2w} = {{\pi\text{/}2} - \chi_{2w}}}}} & (18) \end{matrix}$ When g_(w)(φ_(2w), ψ_(b2w): C₂) is transformed from the coordinate system C₂ to the coordinate system C_(s), g_(w)(ψ₀, φ_(n0): C₃) can be represented as follows: sin φ_(n0)=cos ψ_(b2w) sin Φ_(2w) cos Γ_(s)+sin ψ_(b2w) sin Γ_(s) tan ψ₀=tan φ_(2w) sin Γ_(s)−tan ψ_(b2w) cos Γ_(s)/cos φ_(2w)  (19) With the expressions (17), (18), and (19), g_(w)(ψ_(gw), φ_(nw)) can be represented by g_(w)(ψ₀, φ_(n0): C_(s)). 4.2B Reference Point P_(w), Based on R_(2w), β_(w), ψ_(rw)

As described above at the beginning of section 4.2A, the pitch cones of the pinion and the gear do not have to contact at the reference point P_(w). In this section, a method is described in which the reference point P_(w) is determined on the coordinate system C_(s) without the use of the pitch cone, and by setting the gear reference radius R_(2w), a phase angle β_(w), and a spiral angle ψ_(rw) of the reference point.

The reference point P_(w) is set in the coordinate system C_(s) as follows: P _(w)(u _(cw) ,v _(cw) ,z _(cw) :C _(s)) When P_(w) is represented with the circle radius r_(w) of the relative velocity, and an angle from the u_(c) axis β_(w), in a form of P_(w)(r_(w), β_(w), z_(cw): C_(s)), u _(cw) =r _(w) cos β_(w) v _(cw) =r _(w) sin β_(w) In addition, as the phase angle β_(w) of the reference point and the spiral angle ψ_(rw) are set based on expression (12) which represents a relationship between a radius r_(w) around the instantaneous axis of the reference point P_(w) and the inclination angle ψ_(rw) of the relative velocity, r _(w) =E tan ψ_(rw)×sin(Σ−Γ_(s))sin Γ_(s)/sin Σ u_(cw) and V_(cw) are determined accordingly.

Next, P_(w)(u_(cw), v_(cw), z_(cw): C_(s)) is converted to the coordinate system C₂ of rotation axis of the second gear. This is already described as expression (13). u _(2cw) =−u _(cw) cos Γ_(s) +z _(cw) sin Γ_(s) v _(2cw) =v _(cw) −v _(cs2) z _(2cw) =−u _(cw) sin Γ_(s) −z _(cw) cos Γ_(s)  (13a) Here, as described in section 4.1, v_(cs2)=E tan Γ_(s)/{tan(Σ−Γ_(s))+tan Γ_(s)}. In addition, there is an expression in expression (13) describing: R _(2w) ² =u _(2cw) ² +v _(2cw) ²  (13b) Thus, by setting the gear reference radius R_(2w), z_(cw) is determined based on expressions (13a) and (13b), and the coordinate of the reference point P_(w) in the coordinate system C_(s) is calculated.

Once the design reference point P_(w) is determined, the pinion reference circle radius R_(1w) can also be calculated based on expression (13).

Because the pitch generating line L_(pw) passing at the design reference point P_(w) is determined, the pitch hyperboloid can be determined. Alternatively, it is also possible to determine a design reference cone in which the gear cone angle Γ_(gw) is approximated to be a value around Γ_(S), and the pinion cone angle γ_(pw) is approximated by Σ−Γ_(pw). Although the reference cones share the design reference point P_(w), the reference cones are not in contact with each other. The tip cone angle can be determined similarly to as in section 4.2A.3.

A contact normal g_(w) is set as g_(w)(ψ_(rw), φ_(nrw); C_(rs)) as shown in FIG. 10. The variable φ_(nrw) represents an angle, on the plane S_(nw), between an intersecting line between the plane S_(nw) and the plane S_(pw) and the contact normal g_(w). The contact normal g_(w) can be converted to g_(w)(ψ₀, φ_(n0); C_(s)) as will be described later. Because ψ_(pw) and ψ_(gw) can be determined based on expression (15), the contact normal g_(w) can be set as g_(w)(ψ_(pw), φ_(nw); S_(nw)) similar to section 4.2A.4.

Conversion of the contact normal from the coordinate system C_(rs) to the coordinate system C_(s) will now be described.

(1) A contact normal g_(w)(ψ_(pw), φ_(nrw); C_(rs)) is set.

(2) When the displacement on the contact normal g_(w) is L_(g), the axial direction components of the displacement L_(g) on the coordinate system C_(rs) are: L _(urs) =−L _(g) sin φ_(nrw) L _(vrs) =L _(g) cos φ_(nrw)·cos ψ_(rw) L _(zrs) =L _(g) cos φ_(nrw)·sin ψ_(rw)

(3) The axial direction components of the coordinate system C_(s) are represented with (L_(urs), L_(vrs), L_(zrs)) as: L _(uc) =L _(urs)·cos β_(w) −L _(vrs)·sin β_(w) L _(vc) =L _(urs)·sin β_(w) +L _(vrs)·cos β_(w) L _(zc) =L _(zrs)

(4) Based on these expressions,

$\begin{matrix} {L_{uc} = {{{- \left( {L_{g}\sin\;\varphi_{nrw}} \right)}\cos\;\beta_{w}} - {\left( {L_{g}\cos\;{\varphi_{nrw} \cdot \cos}\;\psi_{rw}} \right)\sin\;\beta_{w}}}} \\ {= {- {L_{g}\left( {{\sin\;{\varphi_{nrw} \cdot \cos}\;\beta_{w}} + {\cos\;{\varphi_{nrw} \cdot \cos}\;{\psi_{rw} \cdot \sin}\;\beta_{w}}} \right)}}} \end{matrix}$ $\begin{matrix} {L_{vc} = {{{- \left( {L_{g}\sin\;\varphi_{nrw}} \right)}\sin\;\beta_{w}} + {\left( {L_{g}\cos\;{\varphi_{nrw} \cdot \cos}\;\psi_{rw}} \right)\cos\;\beta_{w}}}} \\ {= {L_{g}\left( {{{- \sin}\;{\varphi_{nrw} \cdot \sin}\;\beta_{w}} + {\cos\;{\varphi_{nrw} \cdot \cos}\;{\psi_{rw} \cdot \cos}\;\beta_{w}}} \right)}} \end{matrix}$

(5) From FIG. 6, the contact normal g_(w)(Φ₀, φ_(n0); C_(s)) is:

$\begin{matrix} {{\tan\;\psi_{0}} = {L_{zc}\text{/}L_{vc}}} \\ {= {\cos\;{\varphi_{nrw} \cdot \sin}\;\psi_{rw}\text{/}\left( {{{- \sin}\;{\varphi_{rw} \cdot \sin}\;\beta_{w}} + {\cos\;{\varphi_{nrw} \cdot}}} \right.}} \\ \left. {\cos\;{\psi_{rw} \cdot \cos}\;\beta_{w}} \right) \end{matrix}$ $\begin{matrix} {{\sin\;\varphi_{n\; 0}} = {{- L_{uc}}\text{/}L_{g}}} \\ {= {{\sin\;{\varphi_{nrw} \cdot \cos}\;\beta_{w}} + {\cos\;{\varphi_{nrw} \cdot \cos}\;{\psi_{rw} \cdot \sin}\;\beta_{w}}}} \end{matrix}$

(6) From FIG. 11, the contact normal g_(w)(φ_(s0), ψ_(sw0); C_(s)) is:

$\begin{matrix} {{\tan\;\varphi_{\;{s\; 0}}} = {{{- L_{uc}}\text{/}L_{vc}} = {\left( {{\sin\;{\varphi_{nrw} \cdot \cos}\;\beta_{w}} + {\cos\;{\varphi_{nrw} \cdot \cos}\;{\psi_{rw} \cdot \sin}\;\beta_{w}}} \right)\text{/}}}} \\ {\left( {{{- \sin}\;{\varphi_{rw} \cdot \sin}\;\beta_{w}} + {\cos\;{\varphi_{nrw} \cdot \cos}\;{\psi_{rw} \cdot \cos}\;\beta_{w}}} \right)} \end{matrix}$ sin  ψ_(sw 0) = L_(zc)/L_(g) = cos  φ_(nrw) ⋅ cos  ψ_(rw)

The simplest practical method is a method in which the design reference point P_(w) is determined with β_(w) set as β_(w)=0, and reference cones are selected in which the gear cone angle is around Γ_(gw)=Γ_(S) and the pinion cone angle is around γ_(pw)=Σ−Γ_(gw). In this method, because β_(w)=0, the contact normal g_(w) is directly set as g_(w)(ψ₀, φ_(n0); C_(s)).

4.3 Tooth Trace Contact Ratio

4.3.1 General Equation of Tooth Trace Contact Ratio

An contact ratio m_(f) along L_(pw) and an contact ratio m_(fcone) along a direction of a line of intersection (P_(w)P_(gcone) in FIG. 13) between an arbitrary cone surface and S_(w0) are calculated with an arbitrary point P_(w) on g_(w)=g₀ as a reference. The other contact ratios m_(z) and m_(s) are similarly determined.

Because the contact normal g_(w) is represented in the coordinate system C_(s) with g_(w)=g₀(ψ₀, φ_(n0): C_(s)), the point P_(w)(u_(2cw), v_(2cw), z_(2cw): C₂) represented in the coordinate system C₂ is converted into the point P_(w)(q_(2cw), −R_(b2w), z_(2cw): C_(q2)) on the coordinate system C_(q2) in the following manner:

$\begin{matrix} {{q_{2\;{cw}} = {{u_{2\;{cw}}\cos\;\chi_{20}} + {v_{2\;{cw}}\sin\;\chi_{20}}}}\begin{matrix} {R_{b\; 2w} = {{u_{2\;{cw}}\sin\;\chi_{20}} - {v_{2\;{cw}}\cos\;\chi_{20}}}} \\ {= {R_{2\; w}{\cos\left( {\varphi_{20} + ɛ_{2w}} \right)}}} \end{matrix}{\chi_{20} = {{\pi\text{/}2} - \varphi_{20}}}{{\tan\; ɛ_{2w}} = {v_{2\;{cw}}\text{/}u_{2\;{cw}}}}{R_{2w} = \left( {u_{2{cw}}^{2} + v_{2{cw}}^{2}} \right)^{1/2}}} & (20) \end{matrix}$ The inclination angle g₀(φ₂₀, ψ_(b20): C₂) of the contact normal g_(w)=g₀, the inclination angle φ_(s0) of the surface of action S_(w0), and the inclination angle ψ_(sw0) of g₀ (=P₀P_(Gswn)) on S_(w0) (FIGS. 12 and 13) are determined in the following manner: (a) For Cylindrical Gears, Crossed Helical Gears, and Worm Gears tan φ₂₀=tan φ_(n0) cos(Γ_(s)−ψ₀) sin ψ_(b20)=sin φ_(n0) sin(Γ_(s)−ψ₀) tan φ_(s0)=tan φ_(n0) cos ψ₀ tan ψ_(sw0)=tan ψ₀ sin φ_(s0) or sin ψ_(sw0)=sin φ_(n0) sin ψ₀  (20a) (b) For Bevel Gears and Hypoid Gears tan φ₂₀=tan φ_(n0) cos Γ_(s)/cos ψ₀+tan ψ₀ sin Γ_(s) sin ψ_(b20)=sin φ_(n0) sin Γ_(s)−cos ψ_(n0) sin ψ₀ cos Γ_(s) tan φ_(s0)=tan φ_(n0)/cos ψ₀ tan ψ_(sw0)=tan ψ₀ cos φ_(s0)  (20b) The derivation of φ_(s0) and ψ_(sw0) are detailed in, for example, Papers of Japan Society of Mechanical Engineers, Part C, Vol. 70, No. 692, c2004-4, Third Report of Design Theory of Power Transmission Gears.

In the following, a calculation is described in the case where the path of contact coincides with the contact normal g_(w)=g₀. If it is assumed that with every rotation of one pitch P_(w) moves to P_(g), and the tangential plane W translates to W_(g), the movement distance P_(w)P_(g) can be represented as follows (FIG. 11): P _(w) P _(g) =P _(gw) =R _(b2w)(2θ_(2p))cos ψ_(b20)  (21) where P_(gw) represents one pitch on g₀ and 2θ_(2p) represents an angular pitch of the ring gear.

When the intersection between L_(pw) and W_(g) is P_(1w), one pitch P_(fw)=P_(1w)P_(w) on the tooth trace L_(pw) is: P _(fw) =P _(gw)/sin ψ_(sw0)  (22)

The relationship between the internal and external circle radii of the ring gear and the face width of the ring gear is: R _(2t) =R _(2h) −F _(g)/sin Γ_(gw) where R_(2t) and R_(2h) represent internal and external circle radii of the ring gear, respectively, F_(g) represents a gear face width on the pitch cone element, and Γ_(gw) represents a pitch cone angle.

Because the effective length F_(1wp) of the tooth trace is a length of the pitch generating line L_(pw) which is cut by the internal and external circles of the ring gear: F _(1wp)={(R _(2h) ² −v _(2pw) ²)^(1/2)−(R _(2t) ² −v _(2pw) ²)^(1/2)}/sin Γ_(s)  (23) Therefore, the general equation for the tooth trace contact ratio m_(f) would be: m _(f) =F _(1wp) /P _(fw)  (24) 4.3.2 for Cylindrical Gear (FIG. 12)

The pitch generating line L_(pw) coincides with the instantaneous axis (Γ_(s)=0), and P_(w) may be anywhere on L_(pw). Normally, P_(w) is taken at the origin of the coordinate system C_(s), and, thus, P_(w)(u_(cw), v_(cw), z_(cw): C_(s)) and the contact normal g_(w)=g₀ (ψ₀, φ_(n0): C_(s)) can be simplified as follows, based on expressions (20) and (20a): P _(w)(0,0,0:C _(s)),P _(w)(0,−v _(cs2) ,O:C ₂) P ₀(q _(2pw) =−v _(cs2) sin χ₂₀ ,−R _(b2w) =−v _(cs2) cos χ₂₀,0:C _(q2)) φ₂₀=φ_(s0),ψ_(b20)=−ψ_(sw0) tan ψ_(b20)=−tan ψ_(sw0)=−tan ψ₀ sin φ₂₀ In other words, the plane S_(w0) and the plane of action G₂₀ coincide with each other. It should be noted, however, that the planes are viewed from opposite directions from each other.

These values can be substituted into expressions (21) and (22) to determine the tooth trace contact ratio m_(f) with the tooth trace direction pitch P_(fw) and expression (24):

$\begin{matrix} {{{P_{gw} = {{R_{b\; 2w}\left( {2\;\theta_{2p}} \right)}\cos\;\psi_{b\; 20}}}P_{fw} = {{{P_{gw}\text{/}\sin\;\psi_{{sw}\; 0}}} = {{{R_{b\; 2w}\left( {2\;\theta_{2p}} \right)}\text{/}\tan\;\psi_{b\; 20}}}}}\begin{matrix} {m_{f} = {{F_{1\;{wp}}\text{/}P_{fw}} = {F\;\tan\;\psi_{0}\text{/}{R_{2w}\left( {2\;\theta_{2p}} \right)}}}} \\ {= {F\;\tan\;\psi_{0}\text{/}p}} \end{matrix}} & (25) \end{matrix}$ where R_(2w)=R_(b2w)/sin φ₂₀ represents a radius of a ring gear reference cylinder, p=R_(2w)(2θ_(2p)) represents a circular pitch, and F=F_(1wp) represents the effective face width.

The expression (25) is a calculation equation of the tooth trace contact ratio of the cylindrical gear of the related art, which is determined with only p, F, and ψ₀ and which does not depend on φ_(n0). This is a special case, which is only true when Γ_(s)=0, and the plane S_(w0) and the plane of action G₂₀ coincide with each other.

4.3.3 for Bevel Gears and Hypoid Gears

For the bevel gears and the hypoid gears, the plane S_(w0) does not coincide with G₂₀ (S_(w0)≠G₂₀), and thus the tooth trace contact ratio m_(f) depends on φ_(n0), and would differ between the drive-side and the coast-side. Therefore, the tooth trace contact ratio m_(f) of the bevel gear or the hypoid gear cannot be determined with the currently used expression (25). In order to check the cases where the currently used expression (25) can hold, the following conditions (a), (b), and (c) are assumed:

(a) the gear is a bevel gear; therefore, the pitch generating line L_(pw) coincides with the instantaneous axis and the design reference point is P_(w)(0, 0, z_(cw): C_(s));

(b) the gear is a crown gear; therefore, Γ_(s)=π/2; and

(c) the path of contact is on the pitch plane; therefore, φ_(n0)=0.

The expressions (20), and (20b)-(24) can be transformed to yield: φ₂₀=ψ₀,ψ_(b20)=0,φ_(s0)=0,ψ_(sw0)=ψ₀ R _(b2w) =R _(2w) cos φ₂₀ =R _(2w) cos ψ₀ P _(gw) =R _(b2w)(2θ_(2p))cos ψ_(b20) =R _(2w)(2θ_(2p))cos ψ₀ P _(fw) =|P _(gw)/sin ψ_(sw0) =|=|R _(2w)(2θ_(2p))/tan ψ₀| m _(f) =F _(1wp) /P _(fw) =F tan ψ₀ /R _(2w)(2θ_(2p))=F tan ψ₀ /p  (26)

The expression (26) is identical to expression (25). In other words, the currently used expression (25) holds in bevel gears which satisfy the above-described conditions (a), (b), and (c). Therefore,

(1) strictly, the expression cannot be applied to normal bevel gears having Γ_(s) different from π/2 (Γ_(s)≠π/2) and φ_(n0) different from 0 (φ_(n0)≠0); and

(2) in a hypoid gear (E≠0), the crown gear does not exist and ε_(2w) differs from 0 (ε_(2w)≠0).

For these reasons, the tooth trace contact ratios of general bevel gears and hypoid gears must be determined with the general expression (24), not the expression (26).

4.4 Calculation Method of Contact Ratio m_(fcone) Along Line of Intersection of Gear Pitch Cone and Surface of Action S_(w0)

The tooth trace contact ratios of the hypoid gear (Gleason method) is calculated based on the expression (26), with an assumption of a virtual spiral bevel gear of ψ₀=(ψ_(pw)+ψ_(gw))/2 (FIG. 9), and this value is assumed to be sufficiently practical. However, there is no theoretical basis for this assumption. In reality, because the line of intersection of the gear pitch cone and the tooth surface is assumed to be the tooth trace curve, the contact ratio is more properly calculated along the line of intersection of the gear pitch cone and the surface of action S_(w0) in the static coordinate system. In the following, the contact ratio m_(fcone) of the hypoid gear is calculated from this viewpoint.

FIG. 13 shows a line of intersection P_(w)P_(gcone) with an arbitrary cone surface which passes through P_(w) on the surface of action S_(w0). Because P_(w)P_(gcone) is a cone curve, it is not a straight line in a strict sense, but P_(w)P_(gcone) is assumed to be a straight line here because the difference is small. When the line of intersection between the surface of action S_(w0) and the plane v_(2c)=0 is P_(ssw)P_(gswn), the line of intersection P_(ssw)P_(gswn) and an arbitrary cone surface passing through P_(w) have an intersection P_(gcone), which is expressed in the following manner: P _(gcone)(u _(cgcone) ,v _(cs2) ,z _(cgcone) :C _(s)) P _(gcone)(u _(2cgcone),0,z _(c2gcone) :C ₂) where u _(cgcone) =u _(cw)+(v _(cw) −v _(cs2))tan φ_(s0) z _(cgcone)={(v _(cs2) −v _(cw))/cos φ_(s0)} tan ψ_(gcone) +z _(cw) u _(2cgcone) =−u _(cgcone) cos Γ_(s) +z _(cgcone) sin Γ_(s) z _(2cgcone) =−u _(cgcone) sin Γ_(s) −z _(cgcone) cos Γ_(s) ψ_(gcone) represents an inclination angle of P_(w)P_(gcone) from P₀P_(ssw) on S_(w0).

Because P_(gcone) is a point on a cone surface of a cone angle Γ_(gcone) passing through P_(w), the following relationship holds. u _(2cgcone) −R _(2w)=−(z _(2cgcone) −z _(2cw))tan Γ_(gcone)  (27)

When a cone angle Γ_(gcone) is set, ψ_(gcone) can be determined through expression (27). Therefore, one pitch P_(cone) along P_(w)P_(gcone) is: P _(cone) =P _(gw)/cos(ψ_(gcone)−ψ_(sw0))  (28)

The contact length F_(1wpcone) along P_(w)P_(gcone) can be determined in the following manner.

In FIG. 11, if an intersection between P_(w)P_(gcone) and L_(pw0) is P_(ws)(u_(cws), 0, z_(cws): C_(s)) u _(cws) =u _(cw) +v _(cw) tan φ_(s0) z _(cws) =z _(cw)−(v _(cw)/cos φ_(s0))tan ψ_(gcone) If an arbitrary point on the straight line P_(w)P_(gcone) is set as Q(u_(cq), v_(cq), z_(cq): C_(s)) (FIG. 11), u_(cq) and v_(cq) can be represented as functions of z_(cq): v _(cq)={(z _(cq) −z _(cws))/tan ψ_(gcone)} cos φ_(s0) u _(cq) =u _(cws) −v _(cq) tan φ_(s0)

If the point Q is represented in the coordinate system C₂ using expression (13), to result in Q (u_(2cq), v_(2cq), z_(2cq): C₂), the radius R_(2cq) of the point Q is: u _(2cq) =−u _(cq) cos Γ_(s) +z _(cq) sin Γ_(s) v _(2cq) =v _(cq) −v _(cs2) R _(2cq)=√(u _(2cq) ² +v _(2cq) ²)

If the values of z_(cq) where R_(2cq)=R_(2h) and R_(2cq)=R_(2t) are z_(cqh) and z_(cqt), the contact length F_(1wpcone) is: F _(1wpcone)=(z _(cqh) −z _(cqt))/sin ψ_(gcone)  (29)

Therefore, the contact ratio m_(fcone) along P_(w)P_(gcone) is: m _(fcone) =F _(1wpcone) /P _(cone)  (30)

The value of m_(fcone) where ψ_(gcone)→π/2 (expression (30)) is the tooth trace contact ratio m_(f) (expression (24)

5. Examples

Table 1 shows specifications of a hypoid gear designed through the Gleason method. The pitch cone is selected such that the radius of curvature of the tooth trace=cutter radius R_(c)=3.75″. In the following, according to the above-described method, the appropriateness of the present embodiment will be shown with a test result by:

(1) first, designing a hypoid gear having the same pitch cone and the same contact normal as Gleason's and calculating the contact ratio m_(fcone) in the direction of the line of intersection of the pitch cone and the surface of action, and

(2) then, designing a hypoid gear with the same ring gear reference circle radius R_(2w), the same pinion spiral angle ψ_(pw), and the same inclination angle φ_(nw) of the contact normal, in which the tooth trace contact ratio on the drive-side and the coast-side are approximately equal to each other.

5.1 Uniform Coordinate Systems C_(s), C₁, and C₂, Reference Point P_(w) and Pitch Generating Line L_(pw)

When values of a shaft angle Σ=90°, an offset E=28 mm, and a gear ratio i₀=47/19 are set, the intersection C_(s) between the instantaneous axis and the line of centers and the inclination angle Γ_(s) of the instantaneous axis are determined in the following manner with respect to the coordinate systems C₁ and C₂: C _(s)(0,24.067,0:C ₂),C _(s)(0,−3.993,0:C ₁),Γ_(s)=67.989°

Based on Table 1, when values of a ring gear reference circle radius R_(2w)=89.255 mm, a pinion spiral angle ψ_(pw)=46.988°, and a ring gear pitch cone angle Γ_(gw)=62.784° are set, the system of equations based on expressions (13), (14), and (15) would have a solution: r _(w)=9.536 mm,β_(w)=11.10°,z _(cw)=97.021

Therefore, the pitch point P_(w) is: P _(w)(9.358,1.836,97.021:C _(s))

The pitch generating line L_(pw) is determined on the coordinate system C_(s) as a straight line passing through the reference point P_(w) and parallel to the instantaneous axis (Γ_(s)=67.989°).

In the following calculations, the internal and external circle radii of the ring gear, R_(2t)=73.87 and R_(2h)=105 are set to be constants.

5.2 Contact Ratio m_(fconeD) of Tooth Surface D (Represented with Index of D) with Contact Normal g_(wD)

Based on Table 1, when g_(wD) is set with g_(wD)(ψ_(gw)=30.859°, φ_(nwD)=150), g_(wD) can be converted into coordinate systems C_(s) and C₂ with expressions (17), (18), and (19), to yield: g _(wD)(φ_(20D)=48.410,ψ_(b20D)=0.20°:C ₂) g _(wD)(ψ_(0D)=46.19°,φ_(nD)=16.48°:C _(s))

The surface of action S_(wD) can be determined on the coordinate system C_(s) by the pitch generating line L_(pw) and g_(wD). In addition, the intersection P_(Od) between g_(wD) and the plane S_(H) and the radius R_(20D) around the gear axis are, based on expression (16): P _(0D)(10.142,0,95.107:C _(s)),R _(20D)=87.739 mm

The contact ratio m_(fconeD) in the direction of the line of intersection between the pitch cone and the surface of action is determined in the following manner.

The inclination angle φ_(s0D) of the surface of action S_(wD), the inclination angle ψ_(swOD) of g_(wD) on S_(wD), and one pitch P_(gWD) on g_(wD) are determined, based on expressions (20), (20b), and (21), as: φ_(s0D)=23.13°,ψ_(sw0D)=43.79°,P _(gwD)=9.894 (1) When Γ_(gw)=Γ_(gcone)=62.784° is set, based on expressions (27)-(30), ψ_(gcone63D)=74.98°,P _(cone63D)=20.56, F _(1wpcone63D)=34.98,m _(fcone63D)=1.701. (2) When Γ_(gcone)=Γ_(s)=67.989° is set, similarly, g _(cone68D)=−89.99°,P _(cone68D)=14.30, F _(1wpcone68D)=34.70,m _(fcone68D)=2.427. (3) When F_(gcone)=72.0° is set, similarly, ψ_(gcone72D)=78.88°,P _(cone72D)=12.09, F _(1wpcone72D)=36.15,m _(fcone72D)=2.989. 5.3 Contact Ratio m_(fconeC) of Tooth Surface C (Represented with Index C) with Contact Normal g_(wC)

When g_(wC)(ψ_(gw)=30.859θ, φ_(nw)=−27.5°) is set, similar to the tooth surface D, g _(wC)(φ_(20C)=28.68°,ψ_(b20C)=−38.22°:C ₂) g _(wC)(ψ_(0C)=40.15°,φ_(n0C)=−25.61°:C _(s)) P _(0C)(8.206,0,95.473:C _(s)),R _(20C)=88.763 mm

The inclination angle φ_(s0C) of the surface of action S_(wC), the inclination angle ψ_(sw0C) of g_(wC) on S_(wc), and one pitch P_(gwC) on g_(wC) are, based on expressions (20), (20b), and (21): φ_(s0C)=−32.10°,ψ_(sw0C)=35.55°,P _(gwC)=9.086 (1) When Γ_(gw)=Γ_(gcone)=62.784° is set, based on expressions (27)-(30), ψ_(gcone63C)=81.08°,P _(cone63C)=12.971, F _(1wpcone63C)=37.86,m _(fcone63C)=2.919. (2) When Γ_(gcone)=Γ_(s)=67.989° is set, similarly, ψ_(gcone68C)=−89.99°,P _(cone68C)=15.628, F _(1wpcone68C)=34.70,m _(fcone68C)=2.220. (3) When Γ_(gcone)=72° is set, similarly, ψ_(gcone72C)=−82.92°,P _(cone72C)=19.061, F _(1wpcone72C)=33.09,m _(fcone72C)=1.736.

According to the Gleason design method, because Γ_(gw)=Γ_(gcone)=62.784°, the contact ratio along the line of intersection between the pitch cone and the surface of action are m_(fcone63D)=1.70 and m_(fcone63C)=2.92, which is very disadvantageous for the tooth surface D. This calculation result can be considered to be explaining the test result of FIG. 16.

In addition, when the ring gear cone angle Γ_(gcone)=s=67.989°, ψ_(gcone)=−89.99° in both the drive-side and the coast-side. Thus, the line of intersection between the cone surface and the surface of action coincides with the pitch generating line L_(pw), the tooth trace contact ratio of the present invention is achieved, and the contact ratio is approximately equal between the drive-side and the coast-side. Because of this, as shown in FIG. 17, a virtual pitch cone C_(pv) passing through a reference point P_(w) determined on the pitch cone angle of Γ_(gw)=62.784° and having the cone angle of P_(gcone)=67.989° and a pinion virtual cone (not shown) having the cone angle γ_(pcone)=Σ−Γ_(gcone)=22.02° can be defined, and the addendum, addendum angle, dedendum, and dedendum angle of the hypoid gear can be determined according to the following standard expressions of gear design, with reference to the virtual pitch cone. In the gear determined as described above, the tooth trace contact ratio of the present embodiment can be realized along the virtual pitch cone angle. α_(g)=Σδ_(t) ×a _(g)/(a _(g) +a _(p))  (31) a _(g) +a _(p) =h _(k) (action tooth size)  (32) where Σδ_(t) represents a sum of the ring gear addendum angle and the ring gear dedendum angle (which changes depending on the tapered tooth depth), a_(g) represents the ring gear addendum angle, a_(g) represents the ring gear addendum, and a_(p) represents the pinion addendum.

The virtual pitch cones C_(1v) of the ring gear and the pinion defined here do not contact each other, although the cones pass through the reference point P_(w1).

The addendum and the addendum angle are defined as shown in FIGS. 18 and 19. More specifically, the addendum angle α_(g) of a ring gear 100 is a difference between cone angles of a pitch cone 102 and a cone 104 generated by the tooth tip of the ring gear, and the dedendum angle β_(g) is similarly a difference between cone angles of the pitch cone 102 and a cone 106 generated by the tooth root of the ring gear. An addendum a_(g) of the ring gear 100 is a distance between the design reference point P_(w) and the gear tooth tip 104 on a straight line which passes through the design reference point P_(w) and which is orthogonal to the pitch cone 102, and the dedendum b_(g) is similarly a distance between the design reference point P_(w) and the tooth root 106 on the above-described straight line. Similar definitions apply for a pinion 110.

By changing the pitch cone angle such that, for example, Γ_(gw)=72°>Γ_(s), it is possible to design the tooth trace contact ratio to be larger on the tooth surface D and smaller on the tooth surface C. Conversely, by changing the pitch cone angle such that, for example, Γ_(gw)=62.784°<Γ_(s), the tooth trace contact ratio would be smaller on the tooth surface D and larger on the tooth surface C.

A design method by the virtual pitch cone C_(pv) will now be additionally described. FIG. 17 shows a pitch cone C_(p1) having a cone angle Γ_(gw)=62.784° according to the Gleason design method and the design reference point P_(w1). As described above, in the Gleason design method, the drive-side tooth surface is disadvantageous in view of the contact ratio. When, on the other hand, the gear is designed with a pitch cone C_(p2) having a cone angle Γ_(gw)=67.989°, the contact ratio can be improved. The design reference point P_(w2) in this case is a point of contact between the pitch cones of the ring gear and the pinion. In other words, the design reference point is changed from the reference point P_(w1) determined based on the Gleason design method to the reference point P_(w2) so that the design reference point is at the point of contact between pitch cones of the ring gear and the pinion.

As already described, if the surface of action intersects the cone surface having the cone angle of Γ_(gw)=67.989° over the entire face width, the above-described tooth trace contact ratio can be realized. In other words, in FIG. 17, when the element of the cone (virtual pitch cone) passing through the design reference point P_(w1) and having the cone angle of 67.989° exists in the gear tooth surface in the face width of the ring gear, the above-described contact ratio can be realized. In order to realize this, a method may be considered in which the addendum angle and the dedendum angle are changed according to the current method. However, this method cannot be realized due to the following reason.

In order for the cone surface having the cone angle of 67.989° (approximately 68°) and the surface of action to intersect over the entire face width without a change in the pitch cone C_(p1), the ring gear addendum angle α_(g) may be increased so that the tip cone angle Γ_(f) is 680. As shown in FIG. 20, by setting the gear addendum angle α_(g) to 5.216°, the tip cone angle Γ_(f)=68° is realized and the above-described tooth trace contact ratio is achieved along the tooth tip. However, if the tooth is designed according to the standard expressions (31) and (32), almost no dedendum of the ring gear exists, and the pinion would consist mostly of the dedendum.

In this case, the pinion would have negative addendum modification, sufficient effective tooth surface cannot be formed, and the strength of the tooth of the pinion is reduced. Thus, such a configuration cannot be realized.

5.4 Hypoid Gear Specifications and Test Results when Γ_(gw) is Set Γ_(gw)=Γ_(s)=67.989°

Table 2 shows hypoid gear specifications when Γ_(gw) is set Γ_(gw)=Γ_(s)=67.989°. Compared to Table 1, identical ring gear reference circle radius R_(2w)=89.255 mm and pinion spiral angle ψ_(pw)=46.988° are employed, and the ring gear pitch cone angle is changed from Γ_(gw)=62.784° to 67.989°. As a result, P_(w) and Γ_(gw) differ as shown in FIG. 17 and, as will be described below, the other specifications are also different. The pitch cone of the gear is in contact with the pitch cone of the pinion at the reference point P_(w). Design reference point P _(w)(9.607,0.825,96.835:C _(s)) Pinion cone radius R _(1w)=45.449 mm Ring gear pitch cone angle Γ_(gw)=67.989° Pinion pitch cone angle γ_(pw)=21.214° Spiral angle on ring gear pitch plane ψ_(gw)=30.768°

With the pressure angles φ_(nwD) and φ_(nwC) identical to Table 1, if g_(wD)(30.768°, 15°) and g_(wC)(30.768°, −27.5°) are set, the inclination angles would differ, in the static coordinate system C_(s), from g_(wD) and g_(wC) of Table 1: g _(wD)(ψ_(0D)=45.86°,φ_(nOD)=19.43°:C _(s)) g _(wC)(ψ_(0C)=43.17°,φ_(n0C)=−22.99°:C _(s))

The inclination angles of g_(wD) and g_(wC) on the surface of action, and one pitch are: φ_(s0D)=26.86°,ψ_(sw0D)=42.59°,P _(gwD)=9.903 φ_(s0C)=−30.19°,ψ_(sw0C)=39.04°,P _(gwC)=9.094

The tooth trace contact ratios are calculated in the following manner based on expressions (22), (23), and (24): P _(fwD)=14.63,F _(1wpD)=34.70,m _(fD)=2.371  Drive-side: P _(fwC)=14.44,F _(1wpC)=34.70,m _(fC)=2.403  Coast-side:

FIG. 21 shows a test result of the specifications of Table 2, and it can be seen that, based on a comparison with FIG. 16, the transmission error is approximately equal between the drive-side and the coast-side, corresponding to the tooth trace contact ratios.

5.5 Specifications of Hypoid Gear when β_(w)=0

Table 3 shows specifications of a hypoid gear when β_(w) is set to 0 (β_(w)=0) in the method of determining the design reference point P_(w) based on R_(2w), β_(w), and ψ_(rw) described in section 4.2B.

6. Computer Aided Design System

In the above-described design of hypoid gears, the design is aided by a computer aided system (CAD) shown in FIG. 22. The CAD system comprises a computer 3 having a processor 1 and a memory 2, an inputting device 4, an outputting device 5, and an external storage device 6. In the external storage device 6, data is read and written from and to a recording medium. On the recording medium, a gear design program for executing the design method of the hypoid gear as described above is recorded in advance, and the program is read from the recording medium as necessary and executed by the computer.

The program can be briefly described as follows. First, a design request value of the hypoid gear and values of variables for determining a tooth surface are acquired. A pitch cone angle Γ_(gcone) of one gear is provisionally set and used along with the acquired values of the variables, and an contact ratio m_(fconeD) of the drive-side tooth surface and an contact ratio m_(fconeC) of the coast-side tooth surface based on the newly defined tooth trace as described above are calculated. The pitch cone angle Γ_(gcone) is changed and the calculation is repeatedly executed so that these contact ratios become predetermined values. When the contact ratios of the tooth surfaces become predetermined values, the pitch cone angle at this point is set as a design value Γ_(gw), and the specifications of the hypoid gear are calculated. The predetermined value of the contact ratio designates a certain range, and values in the range. Desirably, the range of the contact ratio is greater than or equal to 2.0. The range may be changed between the drive-side and the coast-side. The initial value of the pitch cone angle Γ_(gcone) to be provisionally set is desirably set to the inclination angle Γ_(s) of the instantaneous axis S.

Another program calculates the gear specifications by setting the pitch cone angle Γ_(gw) to the inclination angle Γ_(s) of the instantaneous axis from the first place, and does not re-adjust the pitch cone angle according to the contact ratio. Because it is known that the contact ratios of the tooth surfaces become approximately equal to each other when the pitch cone angle Γ_(gw) is set to the inclination angle Γ_(s), of the instantaneous axis, such a program is sufficient as a simple method.

FIG. 23 shows a gear ratio and a tip cone angle (face angle) Γ_(f) of a uniform tooth depth in which a tooth depth is constant along a face width direction, manufactured through a face hobbing designed by the current method. The uniform tooth depth hypoid gear is a gear in which both the addendum angle α_(g) and the dedendum angle β_(g) of FIG. 19 are 0°, and, consequently, the tip cone angle Γ_(f) is equal to the pitch cone angle Γ_(pw). The specifications of the tooth are determined by setting addendum and dedendum in the uniform tooth depth hypoid gear. In a uniform tooth depth hypoid gear, as shown in FIG. 25, while a cutter revolves around the center axis II of the ring gear 14, the cutter rotates with a cutter center cc as the center of rotation. With this motion, the edge of the cutter moves in an epicycloidal shape and the tooth trace curve is also in an epicycloidal shape. In hypoid gears having a ratio r_(c)/D_(g0) between a cutter radius r_(c) and an outer diameter D_(g0) of less than or equal to 0.52, a ratio E/D_(g0) between an offset E and the outer diameter D_(g0) of greater than or equal to 0.111, and a gear ratio of greater than or equal to 2 and less than or equal to 5, as shown in FIG. 23, hypoid gears around the inclination angle Γ_(s) of the instantaneous axis S are not designed. On the other hand, with the design method of the cone angle according to the present embedment, hypoid gears having the tip cone angle Γ_(f) which is around Γ_(s) can be designed.

FIG. 24 shows a gear ratio and a tip cone angle (face angle) Γ_(f) of a tapered tooth depth gear in which a tooth depth changes along a face width direction, manufactured through a face milling designed by the current method. As shown in FIG. 19, the tip cone angle Γ_(f) is a sum of the pitch cone angle Γ_(gw) and the addendum angle α_(g), and is a value determined by a sum of the ring gear addendum angle and the ring gear dedendum angle, the ring gear addendum, and the pinion addendum, as shown in expression (31). In a tapered tooth depth hypoid gear, as shown in FIG. 26, a radius of curvature of the tooth trace of the ring gear 14 is equal to the cutter radius r_(c). In hypoid gears having a ratio r_(c)/D_(g0) between a cutter radius r_(c) and an outer diameter D_(g0) of less than or equal to 0.52, a ratio E/D_(g0) between an offset E and the outer diameter D_(g0) of greater than or equal to 0.111, and a gear ratio of greater than or equal to 2 and less than or equal to 5, as shown in FIG. 24, hypoid gears of greater than or equal to the inclination angle Γ_(s) of the instantaneous axis S are not designed. On the other hand, the hypoid gear according to the present embedment has a tip cone angle of greater than or equal to Γ_(s) although the variables are within the above-described range, as shown with reference “A” in FIG. 24. Therefore, specifications that depart from the current method are designed.

TABLE 1 PINION RING GEAR SHAFT ANGLE Σ 90°   OFFSET E 28    NUMBER OF TEETH N1, N2 19    47    INCLINATION ANGLE Γ_(s) OF 67.989° INSTANTANEOUS AXIS CUTTER RADIUS R_(c) (RADIUS OF 3.75″ CURVATURE OF GEAR TOOTH TRACE) REFERENCE CIRCLE RADIUS 45.406  89.255  R 1w, R 2w PITCH CONE ANGLE γ pw, Γ gw 26.291° 62.784° SPIRAL ANGLE ON PITCH PLANE 46.988° 30.858° ϕ pw, ϕ gw TIP CONE ANGLE 30.728° 63.713° INTERNAL AND EXTERNAL RADII 73.9, 105, (35) OF GEAR (FACE WIDTH) R 2t, R 2h (Fq) GEAR ADDENDUM 1.22  GEAR DEDENDUM 6.83  GEAR WORKING DEPTH 7.15  CONTACT RATIO (GLEASON METHOD) DRIVE-SIDE COAST-SIDE PRESSURE ANGLE ϕ nw 15°   −27.5°   TRANSVERSE CONTACT RATIO 1.13 0.78 TRACE CONTACT RATIO 2.45 2.45 (NEW CALCULATION METHOD (1.70) (2.92) mfcone)

TABLE 2 PINION RING GEAR SHAFT ANGLE Σ 90°   OFFSET E 28    NUMBER OF TEETH N1, N2 19    47    INCLINATION ANGLE Γ_(s) OF 67.989° INSTANTANEOUS AXIS CUTTER RADIUS R_(c) (RADIUS OF ARBITRARY CURVATURE OF GEAR TOOTH TRACE) REFERENCE CIRCLE RADIUS 45.449  89.255  R 1w, R 2w PITCH CONE ANGLE γ pw, Γ gw 21.214° 67.989° SPIRAL ANGLE ON PITCH PLANE 46.988° 30.768° ϕ pw, ϕ gw TIP CONE ANGLE 25.267° 68.850° INTERNAL AND EXTERNAL RADII 73.9, 105 OF GEAR R 2t, R 2h GEAR ADDENDUM 1.22  GEAR DEDENDUM 6.83  GEAR WORKING DEPTH 7.15  CONTACT RATIO (NEW CALCULATION METHOD) DRIVE-SIDE COAST-SIDE PRESSURE ANGLE ϕ nw 15°   −27.5°   TRANSVERSE CONTACT RATIO 1.05 0.85 ms TOOTH TRACE CONTACT RATIO 2.37 2.40 mf

TABLE 3 PINION RING GEAR SHAFT ANGLE Σ 90° OFFSET E 28  NUMBER OF TEETH N1, N2 19  47    INCLINATION ANGLE Γ_(s) OF    67.989° INSTANTANEOUS AXIS CUTTER RADIUS R_(c) (RADIUS OF ARBITRARY CURVATURE OF GEAR TOOTH TRACE) DESIGN REFERENCE POINT P_(w) (9.73, 0, 96.64) REFERENCE CIRCLE RADIUS  45.41 89.255 R_(1w), R_(2w) PITCH CONE ANGLE γ_(pw), Γ_(gw) 22° 68°   SPIRAL ANGLE ψ_(rw) = ψ₀ 45° TIP CONE ANGLE 22° 68°   INTERNAL AND EXTERNAL RADII 73.9, 105 OF GEAR (FACE WIDTH) R_(2t), R_(2h) GEAR ADDENDUM 1.22 GEAR DEDENDUM 6.83 GEAR WORKING TOOTH DEPTH 7.15 CONTACT RATIO (NEW CALCULATION METHOD) DRIVE-SIDE COAST-SIDE PRESSURE ANGLE φ_(n0D), φ_(n0C) 18°   −20°    TRANSVERSE CONTACT RATIO 1.34 0.63 m_(s) TOOTH TRACE CONTACT RATIO 2.43 2.64 m_(f) 

The invention claimed is:
 1. A hypoid gear comprising a pair of gears including a first gear and a second gear, wherein the hypoid gear is a spiral hypoid gear, and a tip cone angle (Γ_(f)) of the second gear is set at a value which is greater than or equal to an inclination angle (Γ_(s)), of an instantaneous axis (S) which is an axis of a relative angular velocity of the first gear and the second gear with respect to a rotational axis of the second gear, and less than or equal to the inclination angle (Γ_(s)) plus five degrees (Γ_(s))+5°).
 2. A hypoid gear according to claim 1, wherein a ratio (r_(c)/D_(g0)) between a radius of curvature (r_(c)) of a tooth trace of the second gear and an outer diameter (D_(g0)) is less than or equal to 0.52, the outer diameter (D_(g0)) being an outer diameter of the second gear at an outermost edge of the second gear.
 3. A hypoid gear according to claim 2, wherein a ratio (E/D_(g0)) between an offset (E) and the outer diameter (D_(g0)) is greater than or equal to 0.111, the offset (E) being a distance between a rotational axis of the first gear and the rotational axis of the second gear.
 4. A hypoid gear according to claim 1, wherein a gear ratio is greater than or equal to 2 and less than or equal to
 5. 